Riemann surface

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#### Complex geometry
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#### Differential geometry
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[[!include synthetic differential geometry - contents]]
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#### Manifolds and cobordisms
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#Contents#
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## Idea

A Riemann surface is a $1$-dimensional [[algebraic geometry|algebro-geometric]] object with good properties.  The name 'surface' comes from the classical case, which is $1$-dimensional over the [[complex numbers]] and therefore $2$-dimensional over the [[real numbers]].

There are several distinct meaning of what is a Riemann surface, and it can be considered in several generalities. But the main definition by far is the classical one.

## Definition

Classically, a **Riemann surface** is a [[connected space|connected]] complex-$1$-dimensional [[complex manifold]], in the strictest sense of 'manifold'.  In other words, it's a [[Hausdorff space|Hausdorff]] [[second countable space]] $M$ which is locally [[homeomorphism|homeomorphic]] to the [[complex plane]] $\mathbb{C}$ via charts (i.e., [[homeomorphisms]]) $\phi_i:U_i \to V_i$ for $U_i \subset M, V_i \subset \mathbb{C}$ open and such that $\phi_j \circ \phi_i^{-1}: V_i \cap V_j \to V_i \cap V_j$ is [[holomorphic function|holomorphic]].

It is not necessary to make any assumption about whether there exists a countable base for the topology (second countable) or whether it has a countable dense subset (separable). This is because Tibor Rado (1923) proved that all Riemann surfaces, without such prior assumptions, must necessarily have a countable base. Thus for elegance this condition is customarily *not* assumed by specialists in Riemann surfaces.

There are generalizations, e.g., over [[local field]]s in [[rigid analytic geometry]].

## Examples

Evidently an [[open subspace]] of a Riemann surface is a Riemann surface.  In particular, an open subset of $\mathbb{C}$ is a Riemann surface in a natural manner.

The [[Riemann sphere]] $P^1(\mathbb{C}) := \mathbb{C} \cup \{ \infty \}$ or $S^2$ is a Riemann surface with the open sets $U_1 = \mathbb{C}, U_2 = \mathbb{C} - \{0\} \cup \{\infty\}$ and the charts
\[ \phi_1 =z, \;\phi_2 = \frac{1}{z}.\]
The transition map is $\frac{1}{z}$ and thus holomorphic on $U_1 \cap U_2 = \mathbb{C}^*$.

An important example comes from [[analytic continuation]], which we will briefly sketch below.   A **function element** is a pair $(f,V)$ where $f: V \to \mathbb{C}$ is holomorphic and $V \subset \mathbb{C}$ is an open disk.
Two function elements $(f,V), (g,W)$ are said to be **direct analytic continuations** of each other if $V \cap W \neq \emptyset$ and $f \equiv g $ on $V \cap W$.  By piecing together direct analytic continuations on a curve, we can talk about the analytic continuation of a function element along a curve (which may or may not exist, but if it does, it is unique).

Starting with a given function element $\gamma = (f,V)$, we can consider the totality $X$ of all equivalence classes of function elements that can be obtained by continuing $\gamma$ along curves in $\mathbb{C}$.  Then $X$ is actually a Riemann surface.

Indeed, we must first put a topology on $X$.  If $(g,W) \in X$ with $W=D_r(w_0)$ centered at $w_0$, then let a neighborhood of $g$ be given by all function elements $(g_w, W')$ for $w \in W, W' \subset W$; these form a basis for a suitable topology on $X$.  Then the coordinate projections $(g,W) \to w_0$ form appropriate local coordinates.  In fact, there is a globally defined map $X \to \mathbb{C}$, whose image in general will be a proper subset of $\mathbb{C}$.

## Properties

### Basic facts

Since we have local coordinates, we can define a map $f: X \to Y$ of Riemann surfaces to be **holomorphic** or **regular** if it is so in local coordinates.  In particular, we can define a holomorphic complex function as a holomorphic map $f: X \to \mathbb{C}$; for [[meromorphic function|meromorphicity]], this becomes $f: X \to S^2$.

Many of the usual theorems of elementary [[complex analysis]] (that is to say, the local ones) transfer immediately to the case of Riemann surfaces.  For instance, we can locally get a [[Laurent series|Laurent expansion]], etc.

+-- {: .num_theorem}
###### Theorem

Let $f: X \to Y$ be a regular map.  If $X$ is compact and $f$ is nonconstant, then $f$ is surjective and $Y$ compact.

=--

To see this, note that $f(X)$ is [[compact space|compact]], and an open subset by the [[open mapping theorem]], so the result follows by connectedness of $Y$.

### Complexified differentials

Since a Riemann surface $X$ is a $2$-dimensional [[smooth manifold]] in the usual (real) sense, it is possible to do the usual [[differential form|exterior calculus]].  We could consider a 1-form to be a section of the (usual) [[cotangent bundle]] $T^*(X)$, but it is more natural to take the **complexified cotangent bundle** $\mathbb{C} \otimes_{\mathbb{R}} T^*(X)$, which we will in the future just abbreviate $T^*(X)$; this should not be confusing since we will only do this when we talk about complex manifolds.  Sections of this bundle will be called (complex-valued) 1-forms.  Similarly, we do the same for 2-forms.

If $z = x + i y$ is  a local coordinate on $X$, defined say on $U \subset X$, define the (complex) differentials
\[d z = d x + i d y , \;d\bar{z} = d x - i d y.\]
These form a basis for the complexified cotangent space at each point of $U$.  There is also a dual basis
\[ \frac{\partial}{\partial z }  := \frac{1}{2}\left( \frac{\partial}{\partial x} - i \frac{\partial}{\partial y}\right), \;
\frac{\partial}{\partial \bar{z} }  := \frac{1}{2}\left( \frac{\partial}{\partial x} + i \frac{\partial}{\partial y}\right)\]
for the complexified tangent space.

We now claim that we can split the tangent space $T(X) = T^{1,0}(X) + T^{0,1}(X)$, where the former consists of multiples of $\frac{\partial}{\partial z}$ and the latter of multiples of  $\frac{\partial}{\partial \bar{z}}$; clearly a similar thing is possible for the cotangent space.
This is always possible locally, and a holomorphic map preserves the decomposition.
One way to  see the last claim quickly is that given $g: U \to \mathbb{C}$ for $U \subset \mathbb{C}$ open and $0 \in U$ (just for convenience), we can write
\[ g(z) = g(0) + Az + A' \bar{z} + o(|z|) \]
where $A =   \frac{\partial g }{\partial  z }(0), A' = \frac{\partial g }{\partial \bar{z} }(0)$, which we will often abbreviate as $g_z(0), g_{\bar{z}}(0)$.  If $\psi: U' \to U$ is holomorphic and conformal sending $z_0 \in U' \to 0 \in U$, we have
\[ g(\phi(\zeta)) = g(\phi(0)) + A \phi'(z_0)(\zeta-z_0) + A' \overline{ \phi'(z_0)(\zeta-z_0)} + o(|z|); \]
in particular, $\phi$ preserves the decomposition of $T_0(\mathbb{C})$.

Given $f: X \to \mathbb{C}$ smooth, we can consider the projections of the 1-form $df$ onto $T^{1,0}(X)$ and $T^{0,1}(X)$, respectively; these will be called $\partial f, \overline{\partial} f$.  Similarly, we define the corresponding operators on 1-forms: to define $\partial \omega$, first project onto $T^{0,1}(M)$ (the reversal is intentional!) and then apply $d$, and vice versa for $\overline{\partial} \omega$.

In particular, if we write in local coordinates $\omega = u d z + v d\bar{z}$, then
\[ \partial \omega = d( v d \bar{z}) = v_z d z \wedge d\bar{z},\]
and
\[ \overline{\partial} \omega  = d( u d z) = u_{\bar{z}} d\bar{z} \wedge d z.\]
To see this, we have tacitly observed that $d v = v_z d z + v_{\bar{z}} d\bar{z}$.

### Picard group of holomorphic line bundles
 {#PicardGroupOfHolomorphicLineBundles}

The [[Picard group]] of a Riemann surface is the group of [[holomorphic line bundles]] in it. Introductions include ([Bobenko, section 8](#Bobenko)).

See also at _[[Narasimhan–Seshadri theorem]]_ and at _[moduli space of connections -- Flat connections over a torus](moduli+space+of+connections#FlatConnectionsOverATorus)_.

### Central theorems

In the theory of Riemann surfaces, there are several important theorems.  Here are two:

* The [[Riemann-Roch theorem]], which analyzes the vector space of meromorphic functions satisfying certain conditions on zeros and poles;

* The [[uniformization theorem]], which partially classifies Riemann surfaces.

### Homotopy type

A [[compact topological space|compact]] Riemann surface of [[genus]] $g \geq 2$ is a [[homotopy 1-type]]. The [[fundamental groupoid]] is a [[Fuchsian group]].

([MO discussion](http://mathoverflow.net/a/93340/381))

### Branched covers

By the [[Riemann existence theorem]], every connected compact [[Riemann surface]] admits the [[structure]] of a [[branched cover of the Riemann sphere]]. ([MO discussion](http://mathoverflow.net/a/53286/381))

### Function field analogy

[[!include function field analogy -- table]]

## Related concepts

* [[annulus]], [[2-sphere]], [[trinion]]

* [[Teichmüller space]]

* [[moduli space of Riemann surfaces]]

* [[super Riemann surface]]

* [[stable vector bundle]]

* [[elliptic curve]]

* [[worldsheet]]

* [[beta-gamma system]]

* [[quadratic differential]]

## References

Historical references:

* [[Hermann Weyl]], _Die Idee der Riemannschen Fläche_, 1913 (*The concept of a Riemann surface*) (on the book, by Peter Schreiber, 2013: [web](http://mathineurope.eu/en/home/47-information/math-calendar/971-1913-publication-of-the-concept-of-a-riemann-surface-by-hermann-weyl))

* {#Rado25} [[Tibor Radó]], *Über den Begriff der Riemannschen Fläche*, Acta Litt. Sci. Szeged, 2 (101-121), 10 ([pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/rado.pdf), [pdf](http://acta.bibl.u-szeged.hu/13312/1/math_002_101-121.pdf))

  > (proving the [[triangulation theorem]])

Monograph:

* {#Gunning66} [[Robert C. Gunning]], *Lectures on Riemann Surfaces*, Princeton University Press (1966) [[pdf](http://ronan.terpereau.perso.math.cnrs.fr/Master_Class_2023_Dijon/GUNNING_Lectures%20on%20Riemann%20Surfaces.pdf)]

Lecture notes:

* {#Bobenko} [[Alexander Bobenko]], _Compact Riemann Surfaces_ lecture notes ([pdf](http://page.math.tu-berlin.de/~bobenko/Lehre/Skripte/RS.pdf))

*  [[Eberhard Freitag]], _Riemann surfaces -- Sheaf theory, Riemann Surfaces, Automorphic forms_, 2013 ([pdf](http://www.rzuser.uni-heidelberg.de/~t91/skripten/riemfl.pdf))

* {#Wehler20} [[Joachim Wehler]], *Riemann surfaces* 2020 ([pdf](https://www.mathematik.uni-muenchen.de/~wehler/20190530_RiemannSurfacesScript.pdf))

[[!redirects Riemann surfaces]]


Riemann surface via valuations

The following is the special case of the general notion of a [[Riemann surface]] over an arbitrary field due to algebraists in 19th century.

One of the first lessons from Mumford's famous Red Book is the "amazing" correspondence between

* Fields which arise as finite algebraic extensions of the field of rational functions $\mathbb{C}(x)$;

* Compact Riemann surfaces (compact complex manifolds of complex dimension 1, or "curves").

The correspondence goes roughly as follows: to each compact Riemann surface $C$, one may associate the field of [[meromorphic functions]] $Mer(C)$, or holomorphic functions $C \to \mathbb{P}^1(\mathbb{C})$. Moreover, for each such $C$, there exists a finite branched covering

$$\phi\colon C \to \mathbb{P}^1(\mathbb{C})$$

which contravariantly induces a field homomorphism $\mathbb{C}(x) \to Mer(C)$.

In the other direction, to each field $K$ of [[transcendence degree]] 1 over $\mathbb{C}$, there is a Riemann surface whose points may be identified with [[valuation rings]] in $K$. (More precisely, with the _discrete_ valuation rings in $K$. All valuation rings of $K$ are discrete except for $K$ itself, which plays the role of a "generic point".)


Riemann theta function

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#### Theta functions
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#### Complex geometry
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[[!include complex geometry - contents]]
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#Contents#
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## Idea

The _Riemann theta functions_ are a special class of [[theta functions]].
They are [[coordinate]]-expressions for the (essentially unique) [[holomorphic sections]] of certain [[line bundles]] over [[Jacobian varieties]] of [[compact topological space|compact]] [[Riemann surfaces]] $\Sigma$.

As such these are functions of two parameters, a complex modulus $\mathbf{\tau}$ varying in the [[Siegel upper half-space]] and encoding the [[complex structure]] on $\Sigma$, as well as a parameter $\mathbf{z}$ varying in $\mathbb{C}^g$ and encoding a point in the [[Jacobian variety]] $\mathbb{C}^g/\mathbb{Z}^g$. As such the Riemann theta functions are the local coordinate expressions of the covariantly constant sections of the [[Hitchin connection]] on the [[moduli space of Riemann surfaces]] (for [[circle group]] [[gauge group]]) ([Hitchin 90, remark 4.12](#Hitchin90)).

The [[partition function]] of a [[2d CFT]]/[[string]] is naturally a Riemann theta function ([AlvaresGaume-Moore-Vafa 86](#AlvaresGaumeMooreVafa86)).

## Related concepts

* [[Theta characteristic]]

## References

A standard account is

* {#Mumford83} [[David Mumford]], _Tata Lectures on Theta_, Birkhäuser 1983

Review includes

* Wikipedia, _[Riemann theta function](http://en.wikipedia.org/wiki/Theta_function#Riemann_theta_function)_

A review with an eye towards the interpretation of the Riemann theta functions as [[partition functions]] of [[2d CFT]]/[[string]] models is in

* {#AlvaresGaumeMooreVafa86} [[Luis Alvarez-Gaumé]], [[Gregory Moore]], [[Cumrun Vafa]], _Theta functions, modular invariance, and strings_, Communications in Mathematical Physics Volume 106, Number 1 (1986), 1-4 ([Euclid](http://projecteuclid.org/euclid.cmp/1104115581))

Discussion as covariantly constant sections of the [[Hitchin connection]] is in

* {#Hitchin90} [[Nigel Hitchin]], _Flat connections and geometric quantization_, : Comm. Math. Phys. Volume 131, Number 2 (1990), 347-380. ([Euclid](http://projecteuclid.org/euclid.cmp/1104200841))

[[!redirects Riemann theta functions]]


Riemann zeta function

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#### Theta functions
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#### Arithmetic geometry
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#Contents#
* table of contents
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## Idea

The _Riemann zeta function_ is the archetypical example of a [[zeta function]], defined by the formula

$$
  \zeta \colon s \mapsto \underoverset{n = 1}{\infty}{\sum} \frac{1}{n^s}
  \,.
$$

From the point of view of [[arithmetic geometry]] and the [[function field analogy]], the Riemann zeta function is the basic case "over [[F1]]" of a tower of [[zeta functions]] for [[arithmetic curves]] given by more general [[number fields]] -- the [[Dedekind zeta functions]] -- and over [[function fields]] -- the [[Weil zeta function]] -- and for [[complex curves]] -- the Selberg [[zeta function of a Riemann surface]] -- and in another direction for higher dimensional [[arithmetic schemes]] -- the [[arithmetic zeta functions]].

## Properties

### Special values

Some of [[special values of L-functions|special values]] of the Riemann zeta function found (for the non-trivial region of non-positive integers) by [[Leonhard Euler]] in 1734 and 1749 are

| n | -5 | -3 | -1 |  0 | 2 | 4 | 6 | 8 |
|---|----|----|----|----|---|---|---|---|
| $\zeta(n)$ | $-\frac{1}{252}$ |  $\frac{1}{120}$ | $-\frac{1}{12}$ | $-\frac{1}{2}$ | $\frac{\pi^2}{6}$ | $\frac{\pi^4}{90}$ | $\frac{\pi^6}{945}$ | $\frac{\pi^8}{9450}$

where for instance the value $\zeta(-1) = -\frac{1}{12}$ turns out to be the [Euler characteristic of the moduli stack of complex elliptic curves](moduli+stack+of+elliptic+curves#EulerCharacteristic) and as such controls much of [[string theory]].

### The completed zeta function
 {#TheCompletedZetaFunction}

The following slight variant of the actual Riemann zeta function typically exhibits its special properties more explicitly.

+-- {: .num_defn #CompletedRiemannZetaFunction}
###### Definition

The _completed_ Riemann zeta function is

$$
  \hat \zeta(s) \coloneqq \pi^{-s/2}\Gamma(s/2)\zeta(s)
 \,,
$$

where $\Gamma(-)$ denotes the [[Gamma function]].

=--

+-- {: .num_prop #AdelicIntegralIncarnationOfCompletedZeta}
###### Proposition

The completed Riemann zeta function, def. \ref{CompletedRiemannZetaFunction}, is the [[adelic integral]]

$$
  \hat \zeta(s)
   =
  \int_{\mathbb{A}_{\mathbb{Q}}^\times }
    \exp(-S(x))
    \;
    {\vert x\vert}^s
    \;
   d \mu_{\mathbb{A}_{\mathbb{Q}}^\times}(x)
$$

where

* $\exp(-S(-))\colon \mathbb{A}_{\mathbb{Q}}^\times$ denotes the function which sends an [[idele]] $x \in \mathbb{A}_{\mathbb{Q}}^\times$ with canonical components $x = (x_\infty, x_2, \cdots, x_p, \cdots)$

  to the product

  $$
    \exp(-S(x)) \coloneqq \exp(-\pi x_\infty^2)\prod_{p \; prime} \chi_{\mathbb{Z}_p}(x_p)
    \,,
  $$

  where $\chi_{\mathbb{Z}_p}$ denotes the [[characteristic function]] of the [[p-adic integers]] inside the [[ring of adeles]];

  ([Goldfeld-Hundley 11, def. 2.2.5](#GoldfeldHundley11)).

* the [[measure]] is essentially the [[Haar measure]] on the [[idele group]]

  $$
    d \mu_{\mathbb{A}_{\mathbb{Q}}^\times}
    \coloneqq
    \underset{p \leq \infty}{\prod}
    d^\times x_p
  $$

  $$
    d^\times x_p
      \coloneqq
    \left\{
       \array{
           \frac{d x_\infty}{{\vert x_\infty\vert}_\infty} & if \; p = \infty
           \\
           \frac{1}{1-p^{-1}}\frac{d x_p}{{\vert x_p\vert}_p} & if\; p \; finite \; prime
       }
    \right.
  $$

  ([Goldfeld-Hundley 11, def. 2.2.3](#GoldfeldHundley11))

=--

(reviewed e.g. in [Fesenko 08 0.1](#Fesenko08), [Garrett 11, section 1](#Garrett11), [Goldfeld-Hundley 11 (2.2.6)](#GoldfeldHundley11)).

### Relation to the Jacobi theta function
 {#RelationToThetaFunctions}

+-- {: .num_prop }
###### Proposition

The completed zeta function, def. \ref{CompletedRiemannZetaFunction}, is the [[Mellin transform]] of the [[Jacobi theta function]]

$$
  \theta(x)\coloneqq \underset{n \in \mathbb{Z}}{\sum} \exp(- \pi n ^2 x)
$$

in that

$$
  \begin{aligned}
    \hat \zeta(s)
    &=
    \int_0^\infty (\theta(x^2)-1) x^s  \frac{d x}{x}
    \\
    & \stackrel{\tau := x^2}{=}
    \frac{1}{2} \int_0^\infty (\theta(\tau)-1) \tau^{s/2}  \frac{d \tau}{\tau}
  \end{aligned}
 \,.
$$

e.g. ([Fesenko 08, section 0.1](#Fesenko08), [Kowalski, example 2.2.5 ](#Kowalski))

=--

In terms of [[idelic integral]] expression for the complete zeta-function of prop. \ref{AdelicIntegralIncarnationOfCompletedZeta}, this comes out as follows:

We compute the integral $\int_{\mathbb{A}^\times} \exp(-S(\alpha))) |\alpha|^s d\mu_{\mathbb{A}^\times}(\alpha)$ -- as in ([Goldfeld-Hundley 11, pages 47-50](#GoldfeldHundley11)) and the remarks by [[Ivan Fesenko]] in ([Goldfeld-Hundley 11, pages 51-51](#GoldfeldHundley11)).

We decompose by the [strong approximation theorem for ideles](group+of+ideles#StrongApproximationTheorem) the integration domain into the [[idele class group]]

$$
  \mathbb{Q}^\times \backslash \mathbb{A}_{\mathbb{Q}}^\times
  =
  (0,\infty)
  \times
  \underset{p}{\prod} \mathbb{Z}_p^\times
$$

and a factor of the non-zero [[rational numbers]]: so we write

$$
  \alpha=x n
$$

where $x$ runs through representatives of $\mathbb{A}^\times/\mathbb{Q}^\times$ and can be chosen as [[ideles]]  $x=(x_2,x_3,...x_\infty)$ with non-archimedean coordinates being units in $\mathbb{Z}_p$ and $x_\infty$ a [[positive number|positive]] [[real number]], and $n$ is a non-zero [[rational number]].

This way the inner integration is $\int_{\mathbb{Q}^\times} \exp(-S(x n))  d n$. Due to the definition of  $\exp(-S(-))$ in prop. \ref{AdelicIntegralIncarnationOfCompletedZeta}, the integrand here is supported on elements $x n\in \mathbb{Z}_p$ for all $p$, and since  $x_p\in  \mathbb{Z}_p^\times$ we deduce $n\in \mathbb{Z}_p$ for all $p$. Since $\mathbb{Q}$ intersected with all $\mathbb{Z}_p$ is $\mathbb{Z}$ (by the [arithmetic fracture square](fracture+theorem#ArithmeticFractureSquares)),
we have

$$
  \begin{aligned}
     \int_{\mathbb{Q}^\times}\exp(-S(x n))  d n
     & =
     \sum_{n\in \mathbb{Z}^\times} exp(-\pi n^2 x_\infty^2)
     \\
     & = \theta(x_\infty^2)-1
  \end{aligned}
$$

Therefore the full integral becomes $\int_{\mathbb{R}_{\gt 0}}( \theta(y^2)-1) y^s \frac{d y}{y}$ with $y=x_\infty$.

### Functional equation

The [[adelic integral]] representation of prop. \ref{AdelicIntegralIncarnationOfCompletedZeta} directly implies the [[functional equation]]

$$
 \hat \zeta(1-s) = \hat \zeta(s)
$$

of the completed zeta function from the [[functional equation]] of the theta function

$$
  \theta(x^2) x = \theta(x^{-2})
$$

which in turn follows from the [[Poisson summation formula]] (see at _[Jacobi theta function -- Functional equation](Jacobi+theta+function#FunctionalEquation)_).

In terms of the adelic integral expression, the functional equation of the theta function (and of the zeta integral)
corresponds to the analytic duality furnished by Fourier transform on the adelic spaces and its subspaces. (due to [Tate 50](#Tate50), reviewed for instance in [Fesenko 08 0.1](#Fesenko08), [Garrett 11, section 1.9](#Garrett11), [Goldfeld-Hundley 11 theorem 2.2.12](#GoldfeldHundley11))

This [[adelic integral]]-method generalizes to [[Dedekind zeta functions]] for any [[algebraic number field]]. This is due to ([Tate 50](#Tate50)), highlighted in ([Goldfeld-Hundley 11, Remark (1) by Ivan Fesenko](#GoldfeldHundley11)).

### Euler product form

...[[Euler product]]...

$$
  \sum_{n=1}^\infty \frac{1}{n^s}
  =
  \underset{p\; prime}{\prod} \frac{1}{1-p^{-s}}
$$

### Analogs over number fields, function fields and complex curves

The [[function field analogy]] in view of the discussion at _[[zeta function of an elliptic differential operator]]_ says that the Riemann zeta function is [[analogy|analogous]] to the regulated [[functional determinant|functional trace]] of a would-be "[[Dirac operator]] on [[Spec(Z)]]".

[[!include zeta-functions and eta-functions and theta-functions and L-functions -- table]]

Other identifications/analogies of the Riemann zeta function (and more generally the [[Dedekind zeta-function]]) with [[partition functions]] in [[physics]] have been proposed, in particular the _[[Bost-Connes system]]_.

See also at _[[function field analogy]]_.

## Related concepts

* [[Riemann hypothesis]]

## References

Discussion in the context of [[adelic integration]] and [[higher arithmetic geometry]] is in

* {#Tate50} [[John Tate]], _Fourier analysis in number fields, and Hecke's zeta-functions_, Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), Thompson, Washington, D.C., pp. 305–34 1950

* {#Fesenko08} [[Ivan Fesenko]], _Adelic approch to the zeta function of arithmetic schemes in dimension two_, Moscow Math. J. 8 (2008), 273–317 ([pdf](https://www.maths.nottingham.ac.uk/personal/ibf/ada.pdf))

with review including

* {#Garrett11} [[Paul Garrett]], _Iwasawa-Tate on ζ-functions and L-functions_, 2011 ([pdf](http://www-users.math.umn.edu/~garrett/m/mfms/notes_c/Iwasawa-Tate.pdf)

* {#Kowalski} E. Kowalski, first part of _Automorphic forms, L-functions and number theory (March 12–16) Three Introductory lectures_ ([pdf](http://www.math.ethz.ch/~kowalski/lectures.pdf))

* {#GoldfeldHundley11} [[Dorian Goldfeld]], [[Joseph Hundley]], chapter 2 of _Automorphic representations and L-functions for the general linear group_, Cambridge Studies in Advanced Mathematics 129, 2011 ([pdf](https://www.maths.nottingham.ac.uk/personal/ibf/text/gl2.pdf))

Discussion in the context of [[p-adic string theory]]:

* [[An Huang]], [[Bogdan Stoica]], [[Shing-Tung Yau]], _General relativity from $p$-adic strings_ ([arXiv:1901.02013](https://arxiv.org/abs/1901.02013))

[[!redirects Riemann zeta-function]]

[[!redirects complete Riemann zeta function]]
[[!redirects complete Riemann zeta-function]]


Riemann-Hilbert correspondence

#Contents#
* table of contents
{:toc}

## Idea

The most basic form of the Riemann--Hilbert correspondence states that the [[category]] of [[flat connection|flat]] [[vector bundles]] on a suitable [[space]] is equivalent to the category of [[local systems]].

While [[Riemann-Hilbert problem|Hilbert's 21st problem]] has a negative solution, there is a generalized sheaf-theoretical formulation which leads to an equivalence of categories discovered by Mebkhout and a bit later also by [[Kashiwara]].

## Statement

Let $X$ be a smooth algebraic variety over $\mathbb{C}$. Let $D(X^\Zar,\mathcal{D}_{X})$ be the derived category of algebraic [[D-modules]] on $X$ (i.e. quasi-coherent $\mathcal{O}_{X}$-modules with [[flat connection]]), and let $D_{rh}^{b}(X^\Zar,\mathcal{D}_{X})$ be the full subcategory of $D^{b}(X^\Zar,\mathcal{D}_{X})$ consisting of objects whose cohomology sheaves are regular holonomic algebraic D-modules. Let $D(X^{an},\mathbb{C})$ be the derived category of sheaves of $\mathbb{C}$-vector spaces on $X^{an}$, and let $D_{c}^{b}(X^{an},\mathbb{C})$ be the full subcategory of $D^{b}(X^{an},\mathbb{C})$ consisting of objects whose cohomology sheaves are Zariski constructible.

Let $\mathcal{E}$ be an algebraic D-module and let $\mathcal{E}^{\hol}=\mathcal{O}_{X}^{\hol}\otimes_{\mathcal{O}_{X}}\mathcal{E}$. The assignment $\mathcal{E}\to\Omega^{*}(\mathcal{E}^{\mathrm{hol}})$ determines a functor

$$dR: D(X^\Zar,\mathcal{D}_{X})\to D(X^{an},\mathbb{C}).$$

\begin{proposition}(Mebkhout, Kashiwara, Beilinson-Bernstein)

The restriction of the functor $dR$ gives an equivalence of categories

$$D_{rh}^{b}(X^\Zar,\mathcal{D}_{X})\xrightarrow{\sim} D_{c}^{b}(X^{an},\mathbb{C})$$

\end{proposition}

## Use in the geometric Langlands correspondence

The Riemann-Hilbert correspondence allows one to translate the automorphic side of the [[geometric Langlands correspondence]] from the language of [[perverse sheaves]] (which in turn is motivated by functions on $Bun_{G}$, via Grothendieck's functions-to-sheaves dictionary) to the language of [[D-modules]] (see [Frenkel05](#Frenkel05), 3.4 to 3.6).

## p-torsion and p-adic Riemann-Hilbert correspondence

[[Bhargav Bhatt]] and [[Jacob Lurie]] have formulated a version of the Riemann-Hilbert correspondence for finite type schemes over a complete and algebraically closed extension of the [[p-adic numbers]]:

\begin{proposition}([Bhatt21](#Bhatt21), Theorem 5.1)

Let $C$ be a complete and algebraically closed extension of $\mathbb{Q}_{p}$ and let $X$ be of finite type over $\mathcal{O}_{C}$. Then there is a natural exact functor

$$\RH: D_{\cons}^{b}(X_{C},\mathbb{F}_{p})\to D_{\qc}^{b}(X\otimes_{\mathcal{O}_{C}} \mathcal{O}_{C}/p).$$

This functor commutes with proper pushforward, intertwines [[Verdier duality]] and [[Grothendieck duality]] in the almost category, and interacts well with the perverse [[t-structure]].
\end{proposition}

Bhatt and Lurie also have a version with p-adic, instead of p-torsion, coefficients (this time the schemes have to be smooth proper varieties over a finite extension of the p-adic numbers).

\begin{proposition}([Bhatt21](#Bhatt21), Theorem 5.4)

Let $K$ be a finite extension of $\mathbb{Q}_{p}$ and let $X$ be a smooth proper variety over $K$. Then there is a natural exact functor

$$\RH_{\mathcal{D}}: D_{wHT}^{b}(X_{C},\mathbb{Q}_{p})\to DF_{coh}(\mathcal{D}_{X})$$

where $D_{wHT}^{b}(X_{C},\mathbb{Q}_{p})$ is a full subcategory of $D_{cons}^{b}(X_{C},\mathbb{Q}_{p})$ spanned by "weakly Hodge-Tate sheaves" and $DF_{coh}(\mathcal{D}_{X})$ is a suitable derived category of [[D-modules|$\mathcal{D}_{X}$-modules]] equipped with a "good" filtration. This functor commutes with proper pushforward, intertwines [[Verdier duality]] and [[Grothendieck duality]] in the almost category, and interacts well with the perverse [[t-structure]].
\end{proposition}

The construction of Bhatt and Lurie is as follows. Let $K$ be a discretely-valued p-adic field and let $X$ be a smooth projective variety over $K$ with $\mathcal{X}$ its associated rigid analytic space. Let $\mathcal{F}$ be a constructible sheaf over $X$. We say that $\mathcal{F}$ is _weakly de Rham_ if the stalk $\mathcal{F}$ is a weakly de Rham Galois representation. More generally if $\mathcal{F}$ is an object of the derived category $D_{c}^{b}(X,\mathbb{Q}_{p})$ we say that $\mathcal{F}$ is _weakly de Rham_ if its cohomology sheaves are weakly de Rham.

Let $\nu:\mathcal{X}_{proet}\to \mathcal{X}_{et}$ be the projection and let $\mathcal{B}_{dR}$ be the de Rham period sheaf on $\mathcal{X}_{proet}$. If $\mathcal{F}$ is weakly de Rham, we define

$$RH(\mathcal{F}):=R\nu_{*}(\nu^{-1}\mathcal{F}\widehat{\otimes}_{\mathbb{Q}_{p}}\mathcal{B}_{dR}\langle u\rangle)$$

[[Lucas Mann]] has also formulated a version of the p-torsion Riemann-Hilbert correspondence for small v-stacks as part of his proof of mod p Poincare duality for [[rigid analytic spaces]]. This version is stated in terms of $\varphi$-modules, which in his context are simultaneously [[solid modules]] and [[almost modules]] over $\mathcal{O}_{X}^{+}/\pi$ together with an action of the Frobenius $\varphi$, and overconvergent etale $\mathbb{F}_{p}$-sheaves (the corresponding derived $\infty$-categories are denoted by $\mathcal{D}_{\square}^{a}(\mathcal{O}_{X}^{+}/\pi)^{\varphi}$ and $\mathcal{D}_{et}(X,\mathbb{F}_{p})^{oc}$ respectively).

\begin{proposition}([Mann22](#Mann22), Theorem 1.2.7, Theorem 3.9.23)

Let $X$ be a small v-stack over $\mathbb{Z}_{p}$ with pseudouniformizer $\pi$ such that $\pi\vert p$. Then the functor

$$-\otimes\mathcal{O}_{X}^{+a}/\pi:\mathcal{D}_{et}(X,\mathbb{F}_{p})^{oc}\to\mathcal{D}_{\square}^{a}(\mathcal{O}_{X}^{+}/\pi)^{\varphi}$$

is fully faithful and induces an equivalence of categories of perfect objects on both sides.

\end{proposition}

## Related entries

* [[Hilbert's problems]]

* [[flat infinity-connection]]

* [[infinity-local system]]

* [[D-modules]]

## References

* [[Alain Connes]], [[Matilde Marcolli]] _[[Noncommutative Geometry, Quantum Fields and Motives]]_

* Wikipedia, _[Riemann-Hilbert correspondence](http://en.wikipedia.org/wiki/Riemann%E2%80%93Hilbert_correspondence)_

* Z. Mebkhout, _Le formalisme des six opérations de Grothendieck pour les $\mathcal{D}_X$-modules cohérents_, Travaux en Cours __35__. Hermann, Paris, 1989. x+254 pp. [MR90m:32026](http://www.ams.org/mathscinet-getitem?mr=1008245)

* [[Andrea D'Agnolo]], [[Masaki Kashiwara]], _Riemann-Hilbert correspondence for holonomic D-modules_, [arxiv/1311.2374](http://arxiv.org/abs/1311.2374)

> The classical Riemann-Hilbert correspondence establishes an equivalence between the triangulated category of _regular_ holonomic D-modules and that of constructible sheaves. In this paper, we prove a Riemann-Hilbert correspondence for holonomic D-modules which are not necessarily regular. The construction of our target category is based on the theory of ind-sheaves by Kashiwara-Schapira and influenced by Tamarkin's work. Among the main ingredients of our proof is the description of the structure of flat meromorphic connections due to Mochizuki and Kedlaya.

Generalization to [[flat infinity-connections]] via the [[dg-nerve]] and [[iterated integrals]] is discussed in

* [[Jonathan Block]], [[Aaron Smith]], _A Riemann--Hilbert correspondence for infinity local systems_, Adv. Math. 252 (2014) 382--405 [arXiv:0908.2843](https://arxiv.org/abs/0908.2843) [doi](https://doi.org/10.1016/j.aim.2013.11.001)

Another approach to higher Riemann-Hilbert

* J. Chuang, J. Holstein, A. Lazarev, _Maurer-Cartan moduli and theorems of Riemann–Hilbert type_, Appl Categor Struct __29__ (2021) 685-728, [doi](https://doi.org/10.1007/s10485-021-09631-3)

The relation to the [[geometric Langlands correspondence]] is discussed in

* {#Frenkel05} [[Edward Frenkel]], _Lectures on the Langlands Program and Conformal Field Theory_, in _Frontiers in number theory, physics, and geometry II_, Springer Berlin Heidelberg, 2007. 387-533. ([arXiv:hep-th/0512172](http://arxiv.org/abs/hep-th/0512172))

An analogue in [[p-adic geometry]] is discussed in

* [[Jacob Lurie]], _A Riemann--Hilbert Correspondence in p-adic Geometry_, [Felix Klein Lectures 2022](https://www.hcm.uni-bonn.de/events/eventpages/felix-klein-lectures/fkl-2022-lurie/).

* {#Bhatt21}[[Bhargav Bhatt]], _Algebraic geometry in mixed characteristic_,  arXiv:[2112.12010](https://arxiv.org/abs/2112.12010)

* {#Mann22} [[Lucas Mann]], *A p-Adic 6-Functor Formalism in Rigid-Analytic Geometry* [[arXiv:2206.02022](https://arxiv.org/abs/2206.02022)]

[[!redirects Riemann-Hilbert correspondence]]
[[!redirects Riemann–Hilbert correspondence]]
[[!redirects Riemann--Hilbert correspondence]]


Riemann-Hilbert problem

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## Overview

Given an [[ordinary differential equation]] (ODE) with [[meromorphic function|meromorphic]] [[coefficients]], solutions will also in general be multivalued functions; thus associated [[monodromies]] around poles are of importance.

The **21st of [[Hilbert's problems]]** states: _Given a series of points in a complex plane and prescribed monodromies around these points, is there a [[Fuchsian equation|Fuchsian ODE]] with these singularities and monodromies?_

In general the answer is negative as it follows by the counterexample provided by Bolibruh (or Bolibruch, see [wikipedia: Hilbert 21](http://en.wikipedia.org/wiki/Hilbert%27s_twenty-first_problem)).

Nowdays generalizations and refinements of this problem are called the **Riemann-Hilbert problem**. For example techniques of finding the corresponding ODE when it is possible, the closely related [[Riemann-Birkhoff factorization]] (realization of a holomorphic matrix function of a circle as a product of a matrix holomorphic on a neighborhood of closed disk and a function of a matrix holomorphic on a neighborhood of an exterior of the disk including infinity and the circle itself). The correspondence between differential equations and monodromies can in fact be established and is true in general if we understand the data much more generally, using sheaf theory. Namely, that is a rough meaning of so-called **Riemann-Hilbert correspondence** ([wikipedia](http://en.wikipedia.org/wiki/Riemann%E2%80%93Hilbert_correspondence)) in sheaf theory on complex manifolds.

In integrable systems theory (and study of special functions) one often says *Riemann-Hilbert method*.

## Related concepts

* [[Riemann-Hilbert correspondence]]

## References and links

While the short wikipedia article on 21st Hilbert problem is discussing in [nice](http://en.wikipedia.org/wiki/Hilbert%27s_twenty-first_problem) generality, the page titled [Riemann-Hilbert problem](http://en.wikipedia.org/wiki/Riemann%E2%80%93Hilbert_problem), which has more formulas, is in fact explaining only the variants of the Riemann-Birkhoff factorization aspect; it has however a nice list of application areas.

* N. Katz, _An overview of Deligne's work on Hilbert's twenty-first problem_, Proc. of Symp. in Pure Math. __28__, 537-557 (1976).

* O. Babelon, D. Bernard, M. Talon, _Introduction to classical integrable systems_, Cambridge Univ. Press 2003.

* G. D. Birkhoff, _The generalized Riemann problem for linear differential equations and the allied problems for linear difference and q-difference equations_, Proc. Amer. Acad. Arts and Sci. __49__ (1913), 531-568.

* D. V. Anosov, A. A. Bolibruch, _The Riemann-Hilbert problem_, Aspects of Math. E22. Friedr. Vieweg & Sohn, Braunschweig, 1994. x+190 pp.

* Alexander R. Its, _The Riemann-Hilbert problem and integrable systems_,   Notices Amer. Math. Soc.  __50__  (2003),  no. 11, 1389--1400, (survey) [pdf](http://www.ams.org/notices/200311/fea-its.pdf)

* E. Corel, E. Compoint, _Stable flags and the Riemann-Hilbert Problem_, [arxiv/1003.5021](http://arxiv.org/abs/1003.5021)

* R. R. Gontsov, V. A. Poberezhnyi, _Various versions of the Riemann–Hilbert problem for linear differential equations
Russian Mathematical Surveys 63:4 (2008), 603–639 (English bibliography free here: [pdf](http://www.turpion.org/php/reference.phtml/ref_rm4547.pdf?journal_id=rm&paper_id=4547&volume=63&issue=4&type=pdf))

* A.A. Bolibrukh, _The Riemann–Hilbert problem_,
Russian Math. Surveys 45:2 (1990), 1–58; Rus. original А. А. Болибрух, "Проблема Римана–Гильберта", УМН 45:2 (1990), 3–47 ([pdf](http://www.mathnet.ru/php/getFT.phtml?jrnid=rm&paperid=4714&volume=45&year=1990&issue=2&fpage=3&what=fullt&option_lang=eng))

* A. A. Bolibruch, S. Malekand, C. Mitschi, _On the generalized Riemann–Hilbert problem with irregular singularities_, Expo. Math.24 (2006) 235–272

* António F. dos Santos, Pedro F. dos Santos, _[[Lax equation]]s, singularities and Riemann-Hilbert problems_, [arxiv/1010.2933](http://arxiv.org/abs/1010.2933)

* Henryk Żołądek, _The monodromy group_, Monografie Matematyczne 67, Birkhäuser 2006

* Alexander I. Bobenko, Alexander Its, _The asymptotic behaviour of the discrete holomorphic map $Z^a$ via the Riemann-Hilbert method_, [arxiv/1409.2667](http://arxiv.org/abs/1409.2667)

The [[AGT correspondence]] is treated with the help of a Riemann-Hilbert problem in

* G. Vartanov, J. Teschner, _Supersymmetric gauge theories, quantization of moduli spaces of flat connections, and conformal field theory_, [arxiv/1302.3778](http://arxiv.org/abs/1302.3778)

[[!redirects Riemann-Hilbert problem]]
[[!redirects Riemann–Hilbert problem]]
[[!redirects Riemann--Hilbert problem]]
[[!redirects Riemann Hilbert problem]]
[[!redirects Riemann-Hilbert method]]
[[!redirects Hilbert's 21st problem]]


Riemannian cobordism

\tableofcontents

## Idea

A _Riemannian cobordism_ is a [[cobordism]] which, as a [[manifold]] is equipped with the structure of a [[Riemannian manifold]].

A definition of a [[category]] in which Riemannian cobordisms are the [[morphism]]s is described at

* [[bordism categories following Stolz-Teichner]].


Riemannian geometry

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## Idea

Riemannian geometry studies [[smooth manifold]]s that are equipped with a [[Riemannian metric]]: [[Riemannian manifolds]].

Riemannian geometry is hence equivalently the [[Cartan geometry]] for inclusions of the [[orthogonal group]] into the [[Euclidean group]].

## Related concepts

* [[fundamental theorem of Riemannian geometry]]

* [[Nash embedding theorem]]

* [[Laplace operator]], [[wave equation]]

* [[spectral geometry]], [[hearing the shape of a drum]], [[spectral triple]]

* [[pseudo-Riemannian geometry]], [[gravity]], [[spacetimes]]

* [[harmonic map]]

* [[calibrated manifold]]

[[!include curvature in Riemannian geometry -- table]]

$\,$

[[!include local and global geometry - table]]

In [[index theory]]:

* [[Stolz conjecture]]

## References

Named after:

* [[Bernhard Riemann]], *[[Über die Hypothesen, welche der Geometrie zu Grunde liegen]]*, Göttingen (1845) [[doi:10.1007/978-3-642-35121-1](https://doi.org/10.1007/978-3-642-35121-1)]

  Engl. transl: [[William Clifford]]: *[[On the hypotheses which underlie geometry]]*, Nature **VIII** (1873) 183-184 [[doi:10.1007/978-3-319-26042-6](https://doi.org/10.1007/978-3-319-26042-6)]

Monographs:

* [[John M. Lee]], _Riemannian manifolds. An introduction to curvature_. Graduate Texts in Mathematics 176 (1997), Springer.  ISBN: 0-387-98271-X.
Second Edition (retitled): *Introduction to Riemannian Manifolds* (2018), Springer.  ISBN: 978-3-319-91754-2 ([doi:10.1007/978-3-319-91755-9](https://doi.org/10.1007/978-3-319-91755-9))

* {#Chavel} Isaac Chavel, _Riemannian geometry – A modern introduction_ Cambridge University Press (1993)

* {#Berger} Marcel Berger, _A panoramic view of Riemannian geometry_

With an eye towards application in [[mathematical physics]]:

* [[Mikio Nakahara]], Chapter 6 of: _[[Geometry, Topology and Physics]]_, IOP 2003 ([doi:10.1201/9781315275826](https://doi.org/10.1201/9781315275826), pdf)

* [[Gerd Rudolph]], [[Matthias Schmidt]], Chapter 2 of: *Differential Geometry and Mathematical Physics Part II. Fibre Bundles, Topology and Gauge Fields*, Springer 2017 ([doi:10.1007/978-94-024-0959-8](https://link.springer.com/book/10.1007/978-94-024-0959-8))

* [[Jürgen Jost]], *Riemannian Geometry and Geometric Analysis*, Springer (2017) [[doi:10.1007/978-3-319-61860-9](https://doi.org/10.1007/978-3-319-61860-9)]


Riemannian geometry - contents

**[[Riemannian geometry]]**

## Basic definitions

* [[Riemannian manifold]]

  * [[metric]], [[isometry]], [[isometry group]]

* [[moduli space of Riemannian metrics]]

* [[pseudo-Riemannian manifold]]

  * [[Lorentzian manifold]], [[spacetime]]

* [[geodesic]]

  * [[geodesic convexity]]

  * [[geodesic flow]]

* [[Levi-Civita connection]]

  * [[Riemann curvature]], [[torsion of a metric connection]]

## Further concepts

* [[Hodge inner product]], [[Hodge star operator]]

* [[gradient]], [[gradient flow]]

## Theorems

* [[Poincaré conjecture]]-theorem

## Applications

* [[gravity]]

  * [[Einstein-Hilbert action]], [[Einstein equations]], [[supergravity]]

[Edit this sidebar](/nlab/edit/Riemannian+geometry+-+contents)


Riemannian manifold

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## Definition

A _Riemannian manifold_ is a [[smooth manifold]] equipped with a [[Riemannian metric]].

See also [[pseudo-Riemannian manifold]]

## Properties

### As $\mathbb{R}$-Riemannian manifolds

[[!include normed division algebra Riemannian geometry -- table]]

## Examples

* [[Euclidean space]]

* [[round sphere]], [[Riemann sphere]]

* [[Taub-NUT space]], [[ALE space]]

## Related concepts

* To be distinguished from [[Riemann surface]]. In particular, the concepts of 2-dimensional Riemannian manifolds and Riemann surfaces are, while closely related, crucially different.

* There is a refinement of topological [[cobordism category|cobordism categories]] to one of [[Riemannian cobordism]]s.

* See also the [[Myers–Steenrod theorem]].

* For [[infinite-dimensional manifold]]s see also [[orthogonal structure]].

* [[geodesic completeness]]

* [[totally geodesic submanifold]]

* The analog in [[complex geometry]] is the notion of [[Kähler manifold]].

* [[vielbein field]], [[Riemannian metric]], [[affine connection]],

* [[Ricci tensor]], [[scalar curvature]]

* [[Riemannian orbifold]]

  * [[conical singularity]]

* [[hyperbolic manifold]]

* [[analytic torsion]]

* [[Lorentzian manifold]], [[causal manifold]]

* [[Nash embedding theorem]]

## References

* [[John M. Lee]], _Riemannian manifolds. An introduction to curvature_. Graduate Texts in Mathematics 176 (1997), Springer.  ISBN: 0-387-98271-X.
Second Edition (retitled): *Introduction to Riemannian Manifolds* (2018), Springer.  ISBN: 978-3-319-91754-2 ([doi:10.1007/978-3-319-91755-9](https://doi.org/10.1007/978-3-319-91755-9))

[[!redirects Riemannian manifolds]]
[[!redirects Riemannian structure]]


Riemannian metric

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## Definition

### In terms of a 2-tensor

A **Riemannian metric** on a [[smooth manifold]] $M$ is defined as a covariant symmetric 2-[[tensor]] $(., .)_p, p \in M$ -- a section of the symmetrized second tensor power of the [[tangent bundle]] -- such that $(v,v)_p \gt 0$ for all $v \in T_p(M)$.  For convenience, we will write $(v,w)$ for $(v,w)_p$.  In other words, a Riemannian metric is a collection of (positive) inner products on each of the [[tangent space]]s $T_p(M)$ such that if $X,Y$ are (smooth) [[vector field]]s, the function $(X,Y): M \to \mathbb{R}$ defined by taking the inner product at each point, is smooth.  A [[manifold]] together with a Riemannian metric is called a [[Riemannian manifold]].

### In terms of a Vielbein

> for the moment see [[Poincare Lie algebra]] and [[first-order formulation of gravity]]

## Examples

There are several ways to get Riemannian metrics:

1. On $\mathbb{R}^n$, there is a standard Riemannian metric coming from the usual inner product.  More generally, if $g_{i j}: \mathbb{R}^n \to \mathbb{R}$ are smooth functions such that the matrix $(g_{i j}(x))$ is symmetric and positive definite for all $x \in \mathbb{R}^n$, we get a Riemannian metric $\sum_{i,j} g_{i j} d x^i \otimes d x^j$ on $\mathbb{R}^n$, where the sum is to be interpreted as a covariant tensor.

2. Given an immersion $N \to M$, a Riemannian metric on $M$ induces one on $N$ in the natural way, simply by pulling back.  For instance, any surface in $\mathbb{R}^3$ has a Riemannian structure based upon the standard Riemannian structure on $\mathbb{R}^3$---based simply on the usual inner product---and induced on the surface.

3. Given an open covering $U_i$ on $M$, Riemannian metrics $(\cdot, \cdot)_i$ on $U_i$, and a partition of unity $\phi_i$ subordinate to the covering $U_i$, we get a Riemannian metric on $M$ by
\[ (v,w)_p := \sum_i \phi_i(p) (v,w)_{i,p}.\]
Thus, using 1) above, any smooth manifold---which necessarily admits partitions of unity---can be given a Riemannian metric.

## Lengths of Curves

A Riemannian metric allows us to take the length of a curve in a manner resembling the standard case.  Given $v \in T_p(M)$, use the notation $\left \Vert{v} \right \Vert := (v,v) = (v,v)_p$.
If $c: I \to M$ is a smooth curve for $I$ an interval in $\mathbb{R}$, we define
\[ l(c) := \int_I \left \Vert{c'(t)}\right \Vert d t;\]
this is easily checked to be independent of parametrization, just as in the usual case.
Using this, we can make a Riemannian manifold $M$ into a metric space: for $p,q \in M$, let
\[ d(p,q) := \inf_{c \mid c(a)=p,c(b)=q} l(c).\]

The metric on $M$ induces the standard [[topology]] on $M$.
To see this, first note that it is a local question, so we can reduce to the case of $M$ an open ball in euclidean space $\mathbb{R}^n$.  Each tangent vector $v \in T_p(M)$ can be viewed as an element of $\mathbb{R}^n$ in a natural way.  Now let $\left \Vert{\cdot}\right \Vert_{\mathbb{R}^n}$ be the standard norm on $\mathbb{R}^n$.  By continuity, we can find $\delta \gt 0$ by shrinking $M$ if necessary such that for all $v \in T_p(M), p \in K$,
\[  \delta \left \Vert{v}\right \Vert_{\mathbb{R}^n} \leq \left \Vert{v}\right \Vert_p \leq \delta^{-1}   \left \Vert{v}\right \Vert_{\mathbb{R}^n} ;\]
in particular, the lengths of curves in $M$ are necessarily comparable to the usual lengths in $\mathbb{R}^n$.  The result now follows.

## Related concepts

* [[signature of a metric]]

* [[invariant Riemannian metric]]

* [[orthogonal structure]]

* [[Riemannian manifold]], [[pseudo-Riemannian manifold]]

* [[Levi-Civita connection]]

* [[moduli space of Riemannian metrics]]

## References

An introduction in terms of [[synthetic differential geometry]] is in

* [[Gonzalo Reyes]], _Metrics, connections and curvature_ ([pdf](http://po-start.com/reyes/wp-content/uploads/2007/01/metrics.pdf))

[[!redirects Riemannian metrics]]

[[!redirects metric tensor]]
[[!redirects metric tensors]]


Riemannian orbifold

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## Idea

The concept of _Riemannian orbifolds_ is the joint generalization of the concepts of _[[Riemannian manifolds]]_ and _[[orbifolds]]_:

A Riemannian orbifold is an [[orbifold]] equipped with an orbifold [[atlas]] where each [[chart]] $(\widehat{U}_i, G)$ is equipped with a [[Riemannian metric]] such that the [[action]] of $G$ is by [[isometries]], and such that the transition functions from one chart to the other are isometries.

A key aspect is that the orbifold [[singularities]] behave like carrying singular [[curvature]], notably there are [[flat orbifolds]] (also "Euclidean orbifolds", i.e. Riemannian orbifolds with vanishing [[Riemann curvature]] away from the [[singularities]]) whose underlying [[topological spaces]] are [[n-spheres]] (see [below](#FlatCompact2DimensionalOrbifolds)).

Key examples of flat orbifolds are global [[homotopy quotients]] $\mathbb{T}^n \sslash G$ of the [[n-torus]] $\mathbb{T}^n$ equipped with its canonical flat [[Riemannian metric]]. These flat orbifolds are called _toroidal orbifolds_.

## Properties

> under construction

Every flat orbifold whose underlying [[metric space]] is [[connected topological space|connected]] and [[complete metric space|complete]]) is a global quotient of [[Euclidean space]]/[[Cartesian space]] $\mathbb{R}^n$

([Ratcliffe 06, 13.3.10](#Ratcliffe06))

## Examples

### Non-compact orbifolds

Basic examples of non-compact Riemannian orbifolds are [[conical singularities]].

In the flat case these are [[homotopy quotients]] of the form $V\sslash G$ for $G$ a [[finite group]] and $V \in RO(G)$ a [[finite-dimensional vector space|finite-dimensional]] [[orthogonal group|orthogonal]] [[linear representation]] of $G$.

\begin{center}

\end{center}

> graphics grabbed from [Blumenhagen-Lüst-Theisen 13](#BlumenhagenLustTheisen13)

For $V = \mathbb{H}$ equipped with the canonical action of [[finite subgroups of SU(2)]] these are the [[ADE-singularities]].

### Compact flat orbifolds from crystallographic groups
 {#CompactFlatOrbifoldsFromCrystallographicGroups}

+-- {: .num_example #CompactFlatOrbifoldFromCrystallographicGroup}
###### Example
**([[compact topological space|compact]] [[flat orbifolds]] from [[crystallographic groups]])**

Let $E$ be a [[Euclidean space]] and $S \subset Iso(E)$ a [[crystallographic group]] [[action|acting]] on it, with translational [[normal subgroup]] [[lattice (discrete subgroup)|lattice]] $N \subset S$ and corresponding [[point group]] $G = S/N$.

$$
  \array{
    &
    1 && 1
    \\
    & \downarrow && \downarrow
    \\
    {\text{normal subgroup}
    \atop
    \text{lattice of translations}}
    &
    N &\subset& E
    &
    {\text{translation} \atop \text{group}}
    \\
    &
    \big\downarrow && \big\downarrow
    \\
    {\text{crystallographic} \atop \text{group}}
    &
    S &\subset& Iso(E)
    &
    {\text{Euclidean}
    \atop
    \text{isometry group}}
    \\
    &
    \big\downarrow && \big\downarrow
    \\
    {\text{point} \atop \text{group}}
    &
    G &\subset& O(E)
    &
    {\text{orthogonal} \atop \text{group}}
    \\
    &
    \downarrow && \downarrow
    \\
    &
    1 && 1
  }
$$

Then the [[action]] of $G$ on $E$ descends to the [[quotient space]] [[torus]] $E/N$ ([this Prop.](crystallographic+group#InducedPointGroupActionOnTorus))

$$
  \array{
    E
      &\overset{g}{\longrightarrow}&
    E
    \\
    \big\downarrow
    &&
    \big\downarrow
    \\
    E/N
      &\underset{g}{\longrightarrow}&
    E/N
  }
$$

The resulting [[homotopy quotient]] $(E/N)\sslash G$ is a compact flat orbifold.

=--

The following is the class of special cases of
Example \ref{CompactFlatOrbifoldFromCrystallographicGroup} for [[point group]] being the [[involution]]-[[action]] by [[reflection]] at a point:

+-- {: .num_example #CoordinateReflectionOnNTorus}
###### Example
**([[coordinate function|coordinate]] [[reflection]] on [[n-torus]])**

Let $\mathbb{T}^d \coloneqq \mathbb{R}^d / \mathbb{Z}^d$ be the [[n-torus|d-torus]] and consider the [[action]] of the [[cyclic group]] $\mathbb{Z}_2$ by canonical [[coordinate function|coordinate]] [[reflection]]

$$
  \array{
    \mathbb{Z}_2 \times \mathbb{T}^d
    &\longrightarrow&
    \mathbb{T}^d
    \\
    (\sigma, \vec x)
    &\mapsto&
     - \vec x
  }
  \,.
$$

The resulting [[homotopy quotient]] [[orbifold]] $\mathbb{T}^d\sslash\mathbb{Z}_2$ has $2^d$ [[singularities]]/[[fixed points]], namely the points with all coordinates in $\{0\,,\, 1/2\, \mathrm{mod} \mathbb{Z}\}$.

In applications to [[string theory]] orbifolds of the form $\mathbb{R}^{p,1} \times \mathbb{T}^d\sslash \mathbb{Z}_2$ play the role of [[orientifold]] [[spacetimes]] with $2^d$ [[Op-planes]].

=--

### Flat compact 2-dimensional orbifolds
 {#FlatCompact2DimensionalOrbifolds}

In 2 dimensions the [[crystallographic groups]] are the "[[wallpaper groups]]". Hence, as a special case of Example \ref{CompactFlatOrbifoldFromCrystallographicGroup},
the flat [[compact topological space|compact]]  2-dimensional orbifolds may be classified as [[homotopy quotients]] of the [[2-torus]] by [[wallpaper groups]] (for review see e.g. [Guerreiro 09](#Guerreiro09)):

> graphics grabbed from [Bettiol-Derdzinski-Piccione 18](#BettiolDerdzinskiPiccione18)

#### 2-sphere
 {#TwoSphereAsFlatOrbifold}

The [[2-sphere]] arises, in particular, as the [[quotient space]] of the [[torus]] $\mathbb{C}/\mathbb{Z}^2$ by the [[involution]] $z \mapsto -z$, exhibited by the [[Weierstrass elliptic function]] (e.g. [Mukase 04, end of 1.4](Weierstrass+elliptic+function#Mukase04). This realizes the 2-sphere as the underlying coarse space of a [[flat orbifold]] with four singular points, called the _[[pillowcase orbifold]]_.

### Flat compact 3-dimensional orbifolds

#### 3-Sphere

Many flat compact oriented 3-dimensional orbifolds have coarse underlying topological space the [[3-sphere]] ([Dunbar 88](#Dunbar88))

### Flat compact 4-dimensional orbifolds
 {#FlatCompact4dOrbifolds}

#### Kummer surface

The orbifold quotient of the [[4-torus]] by the sign [[involution]] on all four canonical [[coordinates]] is the flat compact 4-dimensional orbifold known as the _[[Kummer surface]]_ $T^4 \sslash \mathbb{Z}_2$ -- the special case of Example \ref{CoordinateReflectionOnNTorus} for $d = 4$. This is a singular [[K3-surface]] (e.g. [Bettiol-Derdzinski-Piccione 18, 5.5](#BettiolDerdzinskiPiccione18))

\begin{center}

\end{center}

> graphics grabbed from [Snowden 11](orbifold#Snowden11)

Also $\mathbb{T}^4\sslash\mathbb{Z}_4$ gives a toroidal orbifold. As [[orientifolds]] with [[D5-branes]] in [[type IIB string theory]] these are discussed in [Buchel-Shiu-Tye 99, Sec. II](#BuchelShiuTye99).

#### 4-Sphere
 {#4SphereAsFlatOrbifold}

The [[complex projective space]] $\mathbb{C}P^2$ is an orbifold quotient of the [[4-torus]] (see [this MO comment](https://mathoverflow.net/a/213009/381)), in generalization of how the [[Weierstrass elliptic function]] exhibits the [[Riemann sphere]] as $\mathbb{T}^2/\mathbb{Z}_2$ ([above](#TwoSphereAsFlatOrbifold)).

A further $\mathbb{Z}_2$-quotient of $\mathbb{C}P^2$ gives the [[4-sphere]] (see at _[4-sphere as quotient of complex projective plane](4-sphere#AsAQuotientOfTheComplexProjectivePlane)_).

\linebreak

### Flat compact 6-dimensional orbifolds

see [FRTV 12](#FRTV12)

### 7-Dimensional $G_2$-orbifolds

see _[[G2-orbifold]]_

## Related concepts

* [[crystallographic group]]

* [[branched cover]]

* [[complex orbifold]]

* [[symplectic orbifold]]

* [[Lorentzian orbifold]]

## References

### General

On [[Riemannian orbifolds]]:

* Joseph Ernest Borzellino, _Riemannian Geometry of Orbifolds_, Dissertation, University of California - Los Angeles, January 1, 1992, pages 1-57.  1992 ([calpoly.edu/math_fac/111](https://digitalcommons.calpoly.edu/math_fac/111))

* {#Ratcliffe06} [[John Ratcliffe]], _Geometric Orbifolds_, chapter 13 in _Foundations of Hyperbolic Manifolds_, Graduate Texts in Mathematics 149, Springer 2006 ([doi:10.1007/978-0-387-47322-2](https://doi.org/10.1007/978-0-387-47322-2), pdf)

* Christian Lange, _Orbifolds from a metric viewpoint_, Geom Dedicata (2020) ([arXiv:1801.03472](https://arxiv.org/abs/1801.03472), [doi:10.1007/s10711-020-00521-x](https://doi.org/10.1007/s10711-020-00521-x))

* {#BettiolDerdzinskiPiccione18} Renato G. Bettiol, Andrzej Derdzinski, [[Paolo Piccione]], _Teichmüller theory and collapse of flat manifolds_, Annali di Matematica (2018) 197: 1247 ([arXiv:1705.08431](https://arxiv.org/abs/1705.08431), [doi:10.1007/s10231-017-0723-7](https://doi.org/10.1007/s10231-017-0723-7))

The [[isometry group]] of [[Riemannian orbifolds]]:

* A. V. Bagaev & N. I. Zhukova, _The isometry groups of Riemannian orbifolds_,  Sib Math J 48, 579–592 (2007) ([doi:10.1007/s11202-007-0060-y](https://doi.org/10.1007/s11202-007-0060-y))

Real-analytic Riemannian orbifolds:

* Marja Kankaanrinta, _On real analytic orbifolds and Riemannian metrics_,
Algebraic & Geometric Topology 13 (2013) 2369–2381 ([doi:10.2140/agt.2013.13.2369](http://dx.doi.org/10.2140/agt.2013.13.2369))

The Yamabe invariant:

* Kazuo Akutagawal, _Computations of the orbifold Yamabe invariant_, Mathematische Zeitschrift volume 271, pages 611–625 (2012) ([doi:10.1007/s00209-011-0880-0](https://doi.org/10.1007/s00209-011-0880-0))

More generally, [[G-structures]] on orbifolds:

* A. V. Bagaev, N. I. Zhukova , _The Automorphism Groups of Finite Type $G$-Structures on Orbifolds_, Siberian Mathematical Journal 44, 213–224 (2003) ([doi:10.1023/A:1022920417785](https://doi.org/10.1023/A:1022920417785))

* [[Robert Wolak]], _Orbifolds, geometric structures and foliations. Applications to harmonic maps_, Rendiconti del seminario matematico - Universita politecnico di Torino, vol. 73/1 , 3-4 (2016), 173-187 ([arXiv:1605.04190](https://arxiv.org/abs/1605.04190))

Discussion of [[gravity]] and maybe [[quantum gravity]] on orbifolds:

* Helio V. Fagundes, Teofilo Vargas, _Orbifolds, Quantum Cosmology, and Nontrivial Topology_ ([arXiv:gr-qc/0611048](https://arxiv.org/abs/gr-qc/0611048))

Discussion of [[perturbative string theory]] on toroidal orbifolds

* {#BlumenhagenLustTheisen13} [[Ralph Blumenhagen]], [[Dieter Lüst]], [[Stefan Theisen]], Chapter 10.5 _Toroidal orbifolds_,  of _Basic Concepts of String Theory_ Part of the series Theoretical and Mathematical Physics pp 585-639 Springer 2013

For more see the references at _[[orbifold]]_.

### Flat orbifolds

* [[John Ratcliffe]], Steven T. Tschantz, _The Calabi construction for compact flat orbifolds_, Topology and its Applications Volume 178, 1 December 2014, Pages 87-106 ([doi:10.1016/j.topol.2014.09.005](https://doi.org/10.1016/j.topol.2014.09.005))

#### Of dimension 2

In 2 dimensions

* [[John Milnor]], _On Lattès Maps_ ([arXiv:math/0402147](https://arxiv.org/abs/math/0402147))

* {#Guerreiro09} João Guerreiro, _Orbifolds and Wallpaper Patterns_ ([pdf](https://faculty.math.illinois.edu/~ruiloja/Estudantes/TrabalhoJGuerreiro.pdf))

2d toroidal [[orientifolds]]:

* Dongfeng Gao, [[Kentaro Hori]], Section 7.3 of: _On The Structure Of The Chan-Paton Factors For D-Branes In Type II Orientifolds_ ([arXiv:1004.3972](https://arxiv.org/abs/1004.3972))

* {#DMR14} [[Charles Doran]], Stefan Mendez-Diez, [[Jonathan Rosenberg]], _String theory on elliptic curve orientifolds and KR-theory_ ([arXiv:1402.4885](http://arxiv.org/abs/1402.4885))

#### Of dimension 3

* {#Dunbar88} [[William Dunbar]], _Geometric orbifolds_, Rev. Mat. Univ. Complutense Madr. 1, No.1-3, 67-99 (1988)

#### Of dimension 4

Flat (toroidal) orbifolds of dimension 4:

In the the context of [[black holes in string theory]]:

* Justin R. David, Gautam Mandal, Spenta R. Wadia, _Microscopic Formulation of Black Holes in String Theory_, Phys.Rept.369:549-686,2002 ([arXiv:hep-th/0203048](https://arxiv.org/abs/hep-th/0203048))

In the context of [[RR-field tadpole cancellation]] for [[intersecting D-brane models]] on toroidal orientifolds:

Specifically [[K3]] [[orientifolds]] ($\mathbb{T}^4/G_{ADE}$) in [[type IIB string theory]], hence for [[D9-branes]] and [[D5-branes]]:

* [[Eric Gimon]], [[Joseph Polchinski]], Section 3.2 of: _Consistency Conditions for Orientifolds and D-Manifolds_, Phys. Rev. D54: 1667-1676, 1996 ([arXiv:hep-th/9601038](https://arxiv.org/abs/hep-th/9601038))

* {#GimonJohnson96} Eric Gimon, [[Clifford Johnson]], _K3 Orientifolds_, Nucl. Phys. B477: 715-745, 1996 ([arXiv:hep-th/9604129](https://arxiv.org/abs/hep-th/9604129))

* {#BuchelShiuTye99} Alex Buchel, [[Gary Shiu]], S.-H. Henry Tye, _Anomaly Cancelations in Orientifolds with Quantized B Flux_, Nucl.Phys. B569 (2000) 329-361 ([arXiv:hep-th/9907203](https://arxiv.org/abs/hep-th/9907203))

* P. Anastasopoulos, A. B. Hammou, _A Classification of Toroidal Orientifold Models_, Nucl. Phys. B729:49-78, 2005 ([arXiv:hep-th/0503044](https://arxiv.org/abs/hep-th/0503044))

Specifically [[K3]] [[orientifolds]] ($\mathbb{T}^4/G_{ADE}$) in [[type IIA string theory]], hence for [[D8-branes]] and [[D4-branes]]:

* {#ParkUranga98} [[Jaemo Park]], [[Angel Uranga]], _A Note on Superconformal N=2 theories and Orientifolds_, Nucl. Phys. B542:139-156, 1999 ([arXiv:hep-th/9808161](https://arxiv.org/abs/hep-th/9808161))

* {#AFIRU00a} G. Aldazabal, S. Franco, [[Luis Ibanez]], R. Rabadan, [[Angel Uranga]], _D=4 Chiral String Compactifications from Intersecting Branes_, J. Math. Phys. 42:3103-3126, 2001 ([arXiv:hep-th/0011073](https://arxiv.org/abs/hep-th/0011073))

* {#AFIRU00b} G. Aldazabal, S. Franco, [[Luis Ibanez]], R. Rabadan, [[Angel Uranga]], _Intersecting Brane Worlds_, JHEP 0102:047, 2001 ([arXiv:hep-ph/0011132](https://arxiv.org/abs/hep-ph/0011132))

* {#KataokaShimojo01} H. Kataoka, M. Shimojo, _$SU(3) \times SU(2) \times U(1)$ Chiral Models from Intersecting D4-/D5-branes_, Progress of Theoretical Physics, Volume 107, Issue 6, June 2002, Pages 1291–1296 ([arXiv:hep-th/0112247](https://arxiv.org/abs/hep-th/0112247), [doi:10.1143/PTP.107.1291](https://doi.org/10.1143/PTP.107.1291))

Systematic construction of [[Ricci tensor|Ricci flat]] [[Riemannian metrics]] on [[K3]] [[orbifolds]]:

* [[Shamit Kachru]], [[Arnav Tripathy]], [[Max Zimet]], _K3 metrics_ ([arXiv:2006.02435](https://arxiv.org/abs/2006.02435))

* [[Arnav Tripathy]], [[Max Zimet]], _A plethora of K3 metrics_ ([arXiv:2010.12581](https://arxiv.org/abs/2010.12581))

In the context of [[Mathieu moonshine]] from [[string]] [[sigma models]] on [[K3]]s:

* [[Matthias Gaberdiel]], [[Stefan Hohenegger]], [[Roberto Volpato]], _Symmetries of K3 sigma models_, Commun.Num.Theor.Phys. 6 (2012) 1-50 ([arXiv:1106.4315](https://arxiv.org/abs/1106.4315))

* [[Matthias Gaberdiel]], Roberto Volpato, _Mathieu Moonshine and Orbifold K3s_ ([arXiv:1206.5143](https://arxiv.org/abs/1206.5143))

#### Of dimension 6

In 6 dimensions (mostly motivated as singular [[supersymmetry and Calabi-Yau manifolds|Calabi-Yau compactifications]] of [[heterotic string theory]] to 4d)

* Jens Erler, [[Albrecht Klemm]], _Comment on the Generation Number in Orbifold Compactifications_, Commun. Math. Phys. 153:579-604, 1993 ([arXiv:hep-th/9207111](https://arxiv.org/abs/hep-th/9207111))

* [[Dieter Lüst]], S. Reffert, E. Scheidegger, S. Stieberger, _Resolved Toroidal Orbifolds and their Orientifolds_, Adv. Theor. Math. Phys. 12:67-183, 2008 ([arXiv:hep-th/0609014](https://arxiv.org/abs/hep-th/0609014))

* S. Reffert, _Toroidal Orbifolds: Resolutions, Orientifolds and Applications in String Phenomenology_, Munich 2006 ([arXiv:hep-th/0609040](https://arxiv.org/abs/hep-th/0609040), [spire:725406](https://inspirehep.net/literature/725406))

* [[Ron Donagi]], [[Katrin Wendland]], _On orbifolds and free fermion constructions_, J. Geom. Phys. 59:942-968, 2009 ([arXiv:0809.0330](https://arxiv.org/abs/0809.0330))

* {#FRTV12} Maximilian Fischer, Michael Ratz, Jesus Torrado, Patrick K.S. Vaudrevange, _Classification of symmetric toroidal orbifolds_, JHEP 1301 (2013) 084 ([arXiv:1209.3906](https://arxiv.org/abs/1209.3906))

[[!redirects Riemannian orbifolds]]

[[!redirects flat orbifold]]
[[!redirects flat orbifolds]]

[[!redirects toroidal orbifold]]
[[!redirects toroidal orbifolds]]


Riesz representation theorem

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Functional analysis
+--{: .hide}
[[!include functional analysis - contents]]
=--
#### Measure and probability theory
+-- {: .hide}
[[!include measure theory - contents]]
=--
=--
=--

# The Riesz representation theorems
* table of contents
{: toc}

## Summary

There are various related theorems in [[functional analysis]] and [[measure theory]] stating, under appropriate conditions, that the [[dual vector space|topological linear duals]] of various familiar [[Banach spaces]] (or something similar) are other familiar Banach spaces.  Most of these are due in part to [[Frigyes Riesz]], and many of them are named after him.  Here we will consider them all together.

Throughout, we use notation for [[integrals]] in which unnecessary '$\mathrm{d}$'s are dropped; see the discussion on notation at _[[measure space]]_.

## $C_c^{{*}{+}} = \overline{RM}^+$

Let $X$ be a [[locally compact Hausdorff space]].  Let $C_c(X)$ be the space of [[continuous functions]] on $X$ (valued in the [[complex numbers]]) with [[compact subset|compact]] [[support]]; make $C_c(X)$ into a [[locally convex space]] with the topology of [[uniform convergence topology|uniform convergence on compact subsets]]; the [[dual vector space]] $C_c(X)*$ of this is (of course) the space of [[continuous map|continuous]] [[linear functional]]s on $C_c(X)$; and the [[positive cone]] $C_c(X)^{{*}{+}}$ of this is the space of [[positive linear functional]]s on $C_c(X)$.  Let $RM(X)$ be the space of [[finite measure|finite]] [[Radon measure]]s on $X$; make $RM(X)$ into a [[Banach space]] with the [[total variation]] norm; the [[extended positive cone]] $\overline{RM(X)}^+$ of this is the space of [[positive measure|positive]] Radon measures on $X$.  [[integral|Integration]] gives a map from $\overline{RM(X)}^+$ to $C_c(X)^{{*}{+}}$:
$$ \mu \mapsto (f \mapsto \int_X f \mu) .$$

+-- {: .num_theorem #Cc}
###### Theorem (Riesz)
This map is a [[homeomorphism]]:
$$ C_c(X)^{{*}{+}} \cong \overline{RM(X)}^+ .$$
=--

## $C_0^* = RM$

Let $X$ be a [[locally compact Hausdorff space]].  Let $C_0(X)$ be the space of [[continuous functions]] on $X$ (valued in the [[complex numbers]]) on the [[one-point compactification]] of $X$ (so vanishing 'at infinity'); make $C_0(X)$ into a [[Banach space]] with the [[supremum norm]].  Let $RM(X)$ be the space of [[finite measure|finite]] [[Radon measure]]s on $X$; make $RM(X)$ into a [[Banach space]] with the [[total variation]] norm.  [[integral|Integration]] gives a map from $RM(X)$ to the [[dual vector space]] $C_0(X)^*$ of $C_0(X)$:
$$ \mu \mapsto (f \mapsto \int_X f \mu) .$$

+-- {: .num_theorem #C0}
###### Theorem (Riesz--Markov)
This map is an [[isometric isomorphism]]:
$$ C_0(X)^* \cong RM(X) .$$
=--

## $L_1^* = L_0$

## $L_p^* = L_{1 - p}$

## $H^* = \bar{H}$

## $L_0^* = BA$

## Related concepts

* [[GNS construction]]

## References

A proof of Theorem \ref{Cc} in [[constructive mathematics]] (in the case where $X$ is a [[compactum]]) is given in

* [[Thierry Coquand]], [[Bas Spitters]], _Integrals and Valuations_ ([arXiv:0808.1522](http://arxiv.org/abs/0808.1522))

[[!redirects Riesz representation theorem]]
[[!redirects Riesz representation theorems]]


Riesz space

## Definition

An __ordered real vector space__ is a real [[vector space]] that has a compatible [[poset]] structure: if $x\le y$, then also $x+a\le y+a$ for all $a$ and $t x\le t y$ for all real $t\ge0$.

A __Riesz space__ is an ordered real vector space whose underlying [[poset]] is a [[lattice]].

A __morphism of Riesz spaces__ is a linear map that preserves finite [[infima]] and [[suprema]].

A __unital Riesz space__ is a Riesz space equipped with a unit: an element $u$ such that $u\ge0$ and for every $x$ there is an integer $n\ge0$ such that $|x|=\sup(x,-x) \le n u$.

A __morphism of unital Riesz spaces__ is a morphism of Riesz spaces that preserves the unit.

Any unit $u$ induces a [[seminorm]]: $\|x\|_u=\inf\{t\mid t\ge0 \; and\; |x|\le t u\}$.

An _Archimedean Riesz space__ is a Riesz space such that any infinitesimal element $e$ is zero.  Here $e$ is __infinitesimal__ if there is $b$ such that $n e\le b$ for all integer $n$.

The seminorm induced by a unit on an Archimedean Riesz space is a [[norm]].

An Archimedean unital Riesz space is __uniformly complete__ if the norm induced by the unit (any unit, in fact) is [[complete]].

## Related concepts

* [[Yosida duality]]

## References

The original definition is in

* [[Frigyes Riesz]], _Sur la décomposition des opérations fonctionelles linéaires_, Atti congress. internaz. mathematici (Bologna, 1928), 3, Zanichelli (1930), 143–148.

[[!redirects Riesz spaces]]
[[!redirects vector lattice]]
[[!redirects vector lattices]]
[[!redirects lattice-ordered vector space]]
[[!redirects lattice-ordered vector spaces]]


Riffs and Rotes

The material previously on this page is now located at the [Online Encyclopedia of Integer Sequences Wiki](https://oeis.org/wiki/Riffs_and_Rotes).

[[!redirects riff]]
[[!redirects rote]]
[[!redirects riffs & rotes]]
[[!redirects Riffs & Rotes]]


Rig

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

[[Rigs]] and rig [[homomorphisms]] form the [[category]] **Rig**.

## Definition

We consider rigs as having an additive [[unit]] $0$, a multiplicative [[unit]] $1$ and being such that $0.x = x.0 = 0$, as discussed in the entry *[[rig]]*.

We recall that a [[rig]] [[homomorphism]] $f \colon R \rightarrow S$ is a [[function]] which is a [[monoid]] [[homomorphism]] for both the additive [[underlying]] monoid and the multiplicative underlying monoid.

## Properties

\begin{proposition}
**(initial object)**
\linebreak
The rig of [[natural numbers]], $\mathbb{N}$, is an [[initial object]] in **Rig**.
\end{proposition}
\begin{proof}
Let $R$ be any rig.
First suppose that we have a [[homomorphism]] $f \colon \mathbb{N} \rightarrow R$. Then $f(n) = n.f(1) = n.1_{R}$. This shows that if such a homomorphism exists, then it is unique.
To show existence, let's verify that we indeed define a rig homomorphism by setting $f(n)=n.1_{R}$. First, we obviously have $f(1)=1_{R}$, $f(0)=0.1_{R}=0_{R}$. Moreover, by [[induction]] on $n, p \,\in\, \mathbb{N}$ we have $f(n+p)=(n+p).1_{R}=(n.1_{R})+(p.1_{R})$ and $f(n p) = n p.1_{R}=(n.1_{R})(p.1_{R})$.
\end{proof}

## Related concepts

* [[Ring]], [[CRing]]

[[!redirects category of rigs]]


rig

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[algebra]], by a _rig_ one means a [[mathematical structure]] much like a [[ring]] but without the assumption that every element has an additive inverse, hence without the assumption of *negatives* (whence the omission of "n" from "ring" [[Schanuel 1991 p. 379](#Schanuel91), [Lawvere 1992 p. 2](#Lawvere92)])

\begin{remark}
**(Terminology: Rigs and semirings)**
Rigs are commonly also called _semirings_, but the term 'semiring' is overloaded in the mathematics literature, with different authors each defining a semiring to be different algebraic structures from each other. See *[[semiring]]* for a discussion about the various definitions of semirings; only one of the proposed definitions is the same as the one of *rigs* as considered here.
\end{remark}

## Definition
 {#Definition}

A _rig_ is a [[set]] $R$ with binary operations of addition and multiplication, such that

* $R$ is a [[monoid]] under multiplication;
* $R$ is a [[commutative monoid]] under addition;
* multiplication distributes over addition, i.e. the distributivity laws hold:
  $$x\cdot (y+z) = (x\cdot y) + (x\cdot z)$$
  $$(y+z)\cdot x = (y\cdot x) + (z\cdot x)$$,
  and also the absorption/annihilation laws, which are their nullary version:
  $$0\cdot x = 0 = x\cdot 0$$

In a ring, absorption follows from distributivity, since $0\cdot x + 0\cdot x = (0+0)\cdot x = 0\cdot x$ and we can cancel one copy to obtain $0\cdot x = 0$.  In a [[rig]], however, we have to assert absorption separately.

More sophisticatedly, we can say that, just as a ring is a [[monoid object]] in [[abelian group]]s, so a rig is a monoid object in [[commutative monoid]]s, where abelian groups and commitative monoids have suitable monoidal structures (they are not the cartesian ones).

Equivalently, a rig is the [[hom-set]] of a category with a single object that is [[enriched category|enriched]] in the category of [[commutative monoids]].

Rigs and rig [[homomorphisms]] form the category [[Rig]].

### Further weakening

As with rings, one sometimes considers non-associative or non-unital versions (where multiplication may not be [[associative]] or may have no [[identity]]).  It is rarer to remove requirements from addition as we have done here.  But notice that while $R$ can be proved (from the other axioms) to be an [[abelian group]] under addition (and therefore a ring) as long as it is a [[group]], this argument does not go through if it is only a monoid.  If we assert only distributivity on one side, however, then we can have a noncommutative addition; see [[near-ring]].

## Properties

Many rigs are either [[ring]]s or [[distributive lattice]]s.  Indeed, a ring is precisely a rig that forms a group under addition, while a distributive lattice is precisely a commutative, simple rig in which both operations are idempotent (see ([Golan 2003, Proposition 2.25](#Golan2003))). Note that a [[Boolean algebra]] is a rig in both ways: as a lattice and as a [[Boolean ring]].

Any rig can be "completed" to a ring by adding negatives, in generalization of how the [[natural numbers]] are completed to the [[integers]].  When applied to the [[set]] of [[isomorphism classes]] of objects in a [[rig category]], the result is part of [[algebraic K-theory]].

More formally, the ring completion of a rig $R$ is obtained by applying the [[group completion]] functor to the underlying additive monoid of $R$, and extending the rig multiplication to a ring multiplication by exploiting distributivity; this gives the [[left adjoint]] $F: Rig \to Ring$ to the forgetful functor $U: Ring \to Rig$. Note however that the unit of the [[adjunction]] $R \to U F(R)$ is not [[monomorphism|monic]] if the additive monoid of $R$ is not [[cancellation monoid|cancellative]], despite an informal convention that "[[completion]]" should usually mean a [[monad]] where the unit *is* monic.

Matrices of rigs can be used to formulate versions of [[matrix mechanics]].

Every rig with [[positive characteristic]] is a [[ring]].

## Examples

Some rigs which are neither rings nor distributive lattices include:

* The [[natural numbers]].
* The nonnegative [[rational number]]s and the nonnegative [[real numbers]].
* Polynomials with coefficients in any rig.
* The set of isomorphism classes of objects in any [[distributive category]], or more generally in any [[rig category]].

* The [[tropical rig]], which is $\mathbb{R}\cup \{\infty\}$ with addition $x\oplus y = min(x,y)$  and multiplication $x\otimes y = x+y$.

  Tropical rigs are among the important class of *[[idempotent semirings]]*.

* The [[ideals]] of a [[commutative ring]] form a rig under ideal addition and multiplication, where the unit and zero ideals are the unit and zero elements of the rig, respectively.  They also form a [[distributive lattice]] and therefore a rig in another way; note that the addition operation is the same in both rigs but the multiplication operation is different (being intersection in the lattice).

## Related concepts

* [[semiring]]

* [[rng]] ([[nonunital ring]])

* [[ring]], [[near-ring]]

* [[tropical geometry]], [[tropical semiring]]

* [[idempotent semiring]]

* A [[categorification]] of the notion of rig is the notion of [[rig category]], or more generally [[colax-distributive rig category]].  See also [[2-rig]] and [[distributivity for monoidal structures]].

* [[semi-monoid]]/[[semi-group]]

* [[semi-category]], [[semi-Segal space]]

* [[semi-simplicial set]]

* [[Burnside rig]]

* [[division rig]]

* [[multiplicatively cancellable rig]]

## References

The terminology "rig" is due to:

* {#Schanuel91} [[Stephen H. Schanuel]], p. 379 of: *Negative sets have Euler characteristic and dimension*, in: *Category Theory*, Lecture Notes in Mathematics **1488** (1991) 379–385 [[doi:10.1007/BFb0084232](https://doi.org/10.1007/BFb0084232)]

* {#Lawvere92} [[William Lawvere]], pp. 1 of: *Introduction to Linear Categories and Applications*, course lecture notes (1992) [[pdf](https://github.com/mattearnshaw/lawvere/blob/192dac273e8bf352f307f87b9ec4fe8ef7dc85b9/pdfs/1992-introduction-to-linear-categories-and-applications.pdf), [[Lawvere-LinearCategories.pdf:file]]]

as recalled in:

* [[F. William Lawvere]]: *The legacy of Steve Schanuel!* (2015) [[web](https://www.acsu.buffalo.edu/~wlawvere/Schanuel%20Memorial%20posting.htm)]

> "We were amused when we finally revealed to each other that we had each independently come up with the term 'rig'."

Discussion under the name *semirings*:

* {#Golan1999} Jonathan S. Golan, _Semirings and their applications_. Updated and expanded version of The theory of semirings, with applications to mathematics and theoretical computer science, Longman Sci. Tech., Harlow, 1992, MR1163371. Kluwer Academic Publishers, Dordrecht, 1999. xii+381 pp.

* {#Golan2003} Jonathan S. Golan, _Semirings and affine equations over them: theory and applications_ (Vol. 556). Springer Science & Business Media, 2003.

* [[M. Marcolli]], R. Thomgren, _Thermodynamical semirings_, [arXiv/1108.2874](http://arxiv.org/abs/1108.2874)

* wikipedia [semiring](http://en.wikipedia.org/wiki/Semiring)

* J. Jun, S. Ray, J. Tolliver, _Lattices, spectral spaces, and closure operations on idempotent semirings_, [arxiv/2001.00808](https://arxiv.org/abs/2001.00808)

[[!redirects rigs]]
[[!redirects Rigs]]
[[!redirects commutative rig]]


rig category

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# Rig categories
* table of contents
{: toc}

## Idea

Recall that a [[rig]] is a '[[ring]] without negatives': a [[monoid object]] in the [[monoidal category]] of [[commutative monoids]] with the usual [[tensor product]].  [[categorification|Categorifying]] this notion, we obtain various notions of _[[2-rig]]_.  One of these, in which both "addition" and "multiplication" are represented by abstract [[monoidal category|monoidal structures]], is the notion of *rig category*, also known as a *bimonoidal category*.

A typical example would be the [[groupoid]] of [[finite set|finite sets]] and [[bijection|bijections]], with [[disjoint union]] playing the role of addition and [[cartesian product]] playing the role of multiplication.  This rig category can be thought of as a categorification of the set of [[natural numbers]].  Note that in this example, disjoint union is _not_ the categorical [[coproduct]], and product of sets is _not_ the categorical [[product]] (because we are working in the *groupoid* of finite sets).

## Definition

A **rig category**, or **bimonoidal category**, $C$ is a category with a [[symmetric monoidal category|symmetric monoidal]] structure $(C,\oplus,0)$ for addition and a [[monoidal category|monoidal]] structure $(C, \otimes, I)$ for multiplication, together with left and right [[distributivity law|distributivity]] [[natural isomorphisms]]

$$
  d_\ell
  \;\colon\;
  x \otimes (y \oplus z)
  \longrightarrow
  (x \otimes y) \oplus (x \otimes z)
$$

$$
  d_r
  \;\colon\;
  (x \oplus y) \otimes z
  \longrightarrow
  (x \otimes z) \oplus (y \otimes z)
$$

and absorption/annihilation [[natural isomorphisms]]

$$ a_\ell \;\colon\; x \otimes 0 \longrightarrow 0 $$

$$ a_r \;\colon\; 0 \otimes x \longrightarrow 0 $$

satisfying a set of [[coherence laws]] worked out in ([Laplaza 72](#Laplaza72)) and ([Kelly74](#Kelly74)).

Note that these authors used the term "ring category".  We take the liberty of switching to "rig category"(as in *[[rig]]*) because it is typical for these to lack additive inverses.

While a rig can have the [[extra property]] of being [[commutative ring|commutative]] (i.e. of its multiplication being commutative), a rig category can have the [[extra structure]] of (its monoidal structure $\otimes$) being [[braided monoidal category|braided]] (compatibly with the distributive laws) and may then have the further property of being [[symmetric monoidal category|symmetric]].

## Examples

Rig categories are part of the hierarchy of [[distributivity for monoidal structures]].

### Distributive categories

A rig category where $\oplus$ is the [[category theory|category-theoretic]] [[coproduct]] and $\otimes$ is the category-theoretic [[product]] ([[Cartesian product]]) is called a _[[distributive category]]_.

For example,

* the category [[Set]] of [[sets]],

* any [[topos]],

* the category [[Top]] of [[topological spaces]] with respect to forming [[product topological spaces]] and [[disjoint union topological spaces]];

are distributive categories, hence rig categories with $\times$ and $+$.

### Distributive monoidal categories
 {#DistributiveMonoidalCategories}

In between, we have the notion of [[distributive monoidal category]], where $\oplus$ is the coproduct but $\otimes$ is a possibly non-cartesian [[monoidal category|monoidal structure]].

Examples of this sort include [[Ab]], $R$[[Mod]], [[Vect]] and [[Vect(X)]]:

* [[Ab]], the category of [[abelian groups]] equipped with the [[tensor product of abelian groups]]

* $R$[[Mod]], the category of [[modules]] over a [[commutative ring]] $R$, equipped  with the [[tensor product of modules]]

* $k$[[Vect]] = $k$[[Mod]], the category of [[vector space]] over some [[field]] $k$, equipped with the [[tensor product of vector spaces]],

* $k$[[Vect(X)]], the category of ([[topological vector bundles|topological]]) [[vector bundles]] for $X$ some ([[topological space|topological]]) [[space]], equipped with the [[tensor product of vector bundles]].

In all these cases the coproduct is the respective [[direct sum]] (e.g. [[direct sum of vector bundles]] in the last case).

Also:

* the category of [[pointed sets]] with respect to forming [[smash product]] and [[wedge sum]],

* more generally, the category of [[pointed topological spaces]] with respect to forming [[smash product]] and [[wedge sum]] (e.g. [Hatcher, Section 4.F](algebraic+topology#Hatcher)).

## Baez's conjecture

\begin{conjecture}
(John Baez)
\linebreak
Using the correct definition of the 2-category of symmetric rig categories, the groupoid $FinSet^{\times}$ of finite sets and bijections is the [[initial object|initial]] symmetric rig category, just as $\N$ is the initial commutative rig.   Note that a suitably weakened concept of 'initial' is needed here; see [[2-limit]].  In other words, given any symmetric rig category $R$, there is a unique symmetric rig morphism $FinSet^{\times} \to R$, up to an [[equivalence]] which is itself unique up to an [[isomorphism]] which is actually unique (up to [[equality]]).
\end{conjecture}

This conjecture was established in ([Elgueta 2021](#Elg21)). See also the proof using sheet diagrams in ([Comfort-Delpeuch-Hedges, Sec. 8](#CDH)), and the detailed proof of a restricted version in ([Johnson-Yau, Part I Sec. 2.7](#JY)).
Yau has also proved a braided analogue of this conjecture ([Johnson-Yau, Part II Thm. 7.3.4](#JY)).

## Related concepts

* [[distributivity for monoidal structures]]

  * [[distributive category]], [[distributive monoidal category]]

  * If the distributivity morphisms are not invertible, then we have the notion of [[colax-distributive rig category]].

  * [[duoidal category]]

  * [[2-rig]]

* [[ring groupoid]]

* The notion of a [[bipermutative category]] is a strictification of the notion of [[symmetric monoidal category|symmetric]] rig category.  Every symmetric rig category is equivalent to a bipermutative category ([May, prop. VI 3.5]).  Similarly, every rig category is equivalent to a strict rig category ([Guillou, theorem 1.2](#Guillou)).

* [[K-theory of a bipermutative category]]

* [[BDR 2-vector bundle]].

## References

A textbook treatment is given in

* {#JY} [[Niles Johnson]], [[Donald Yau]], _Bimonoidal Categories, $E_n$-Monoidal Categories, and Algebraic K-Theory_ ([three volume book](https://nilesjohnson.net/En-monoidal.html); abridged version [arXiv:2107.10526](https://arxiv.org/abs/2107.10526)).

The [[coherence law|coherence]] for the [[distributivity law]] in bimonoidal categories has been given in

* {#Laplaza72} [[M. L. Laplaza]], _Coherence for distributivity_, Lecture Notes in Mathematics **281** Springer (1972) 29-72 [[pdf](https://people.math.rochester.edu/faculty/doug/otherpapers/laplaza.pdf)]

* {#Kelly74} [[G. M. Kelly]], _Coherence theorems for lax algebras and distributive laws_, Lecture Notes in Mathematics 420, Springer Verlag, Berlin, 1974, pp. 281-375.

where these categories are called _ring categories_. Discussion with an eye towards the [[K-theory of a bipermutative category]] is in

* [[Peter May]], _$E_\infty$ Ring Spaces and $E_\infty$ Ring spectra_, Springer lectures notes in mathematics, Vol. 533, (1977) ([pdf](http://www.math.uchicago.edu/~may/BOOKS/e_infty.pdf)) chaper VI
 {#May}

* Bertrand Guillou, _Strictification of categories weakly enriched in symmetric monoidal categories, [arXiv:0909.5270](http://arxiv.org/abs/0909.5270)
 {#Guillou}

* Angélica Osorno, _Spectra associated to symmetric monoidal bicategories_ ([arXiv:1005.2227](http://arxiv.org/abs/1005.2227))

A [[string diagram]] treatment of rig categories via _sheet diagrams_ is in

* {#CDH} Cole Comfort, Antonin Delpeuch, [[Jules Hedges]], _Sheet diagrams for bimonoidal categories_, ([arXiv:2010.13361](https://arxiv.org/abs/2010.13361))

[[bi-initial|Biinitiality]] of the groupoid of finite sets is shown in

* {#Elg21} Josep Elgueta, _The groupoid of finite sets is biinitial in the 2-category of rig categories_, Journal of Pure and Applied Algebra
**225** Issue 11 (2021) 106738, doi:[10.1016/j.jpaa.2021.106738](https://doi.org/10.1016/j.jpaa.2021.106738), [arXiv:2004.08684](https://arxiv.org/abs/2004.08684).

On the idea of a [[symmetric monoidal category]] as a [[module]] over a commutative rig category:

* [[John D. Berman]], _On the commutative algebra of categories_, Algebraic and Geometric Topology, 18(5):2963–3012, August 2018, ([arXiv:1606.05606](https://arxiv.org/abs/1606.05606)).

[[!redirects rig category]]
[[!redirects rig categories]]

[[!redirects rig-category]]
[[!redirects rig-categories]]

[[!redirects semiring category]]
[[!redirects semiring categories]]

[[!redirects ring category]]
[[!redirects ring categories]]
[[!redirects bimonoidal category]]
[[!redirects bimonoidal categories]]


rigged Hilbert space

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## Idea

The notion of _rigged Hilbert spaces_ serves to treat [[spectral theory]] of normal [[unbounded operators]] on a [[Hilbert space]] much as if it were about actual [[eigenvalues]] and [[eigenvectors]]. It may be used to make rigorous the idea of [[eigenstates]] for [[quantum observables]] as used in [[quantum mechanics]].

## Example

Let $H$ be the [[Hilbert space]] $L^2(\mathbb{R}, d x)$ consisting of square integrable functions $f$ with respect to [[Lebesgue measure]]. There is an [[unbounded operator|unbounded]] [[self-adjoint operator]]

$$
  \array{
    m_x \,\colon\,
    &
    H &\longrightarrow& H
    \\
    & f &\mapsto& x \cdot f
    \mathrlap{\,,}
  }
$$

where

$$
  (x \cdot f)(y) \coloneqq y f(y)
  \,.
$$

This operator is not defined on all of $H$, but it is defined on a [[dense subspace]] of $H$. For example, if $S$ is the [[Schwartz space]] consisting of [[smooth functions]] $f$ on $\mathbb{R}$ all of whose [[derivatives]] $f^{(n)}(x)$ [[rapidly decreasing function|decay rapidly]] at infinity (more rapidly than any negative power of $|x|$), then there is a dense inclusion map $i: S \to H$, and $m_x$ is defined globally on $S$.

Meanwhile the Schwartz space $S$ carries its own topology (as described in the article *[[distribution]]*), stronger than the topology it inherits from $H$, and the space of tempered distributions $S^*$ is defined to be the continuous dual of the [[topological vector space]] $S$. Since the continuous inclusion $i: S \to H$ is dense, it follows that any continuous functional

$$f: S \to \mathbb{C}$$

has at most one extension to a continuous functional $H \to \mathbb{C}$. In other words, the adjoint map

$$i^*: H^* \to S^*$$

is injective. In addition, the topology on $S$ is such that the operator $m_x: S \to S$ is continuous.

In this example, there is a dense inclusion $S \to S^*$ defined by the inner product pairing, and the operator $m_x$ extends uniquely to an operator $S^* \to S^*$, called $m_x$ by abuse of notation. Again, in this example, the operator $m_x: S^* \to S^*$ has an eigenvector $s_{\xi}$ for each $\xi \in \mathbb{R}$:

$$m_x(s_{\xi}) = \xi s_{\xi}$$

_Ugh. Lousy start on something that would be nice to understand properly. Maybe an expert can help out. John, you there?_

## References

Among the original treatises on the theory of rigged Hilbert spaces is

* [[John Roberts]], _Rigged Hilbert spaces in quantum mechanics_, Communications in Mathematical Physics, *3*  (1966)  98–119. doi:[10.1007/BF01645448](https://doi.org/10.1007/BF01645448)

A unification of various inequivalent approaches is claimed to be achieved in

* M. Gadella, F. Gómez, _A Unified Mathematical Formalism for the Dirac Formulation of Quantum Mechanics_, Foundations of Physics *32* (2002) 815–869, doi:[10.1023/A:1016069311589](https://doi.org/10.1023/A:1016069311589)

See also

* S. Wickramasekara, A. Bohm, _Symmetry Representations in the Rigged Hilbert Space Formulation of Quantum Mechanics_ ([arXiv](http://arxiv.org/abs/math-ph/0302018))

Rigged Hilbert spaces are also known as [[Gelfand triple]]s, written about in his series on generalized functions

* Gel'fand, I. M., and Shilov, Georgiĭ Evgenʹevich. Generalized Functions. United States, American Mathematical Society, (2016)

## See also

Discussion on MathOverflow

* [_Good references for Rigged Hilbert spaces?_](https://mathoverflow.net/questions/43313/good-references-for-rigged-hilbert-spaces)

[[!redirects rigged Hilbert spaces]]


rigged limit

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# Rigged limits
* table of contents
{: toc}

## Idea

A **rigged limit** is a [[2-limit]] which is [[created limit|created]] in 2-categories of algebras and [[lax morphisms|lax]], colax, or pseudo morphisms for a 2-monad.

In order to characterize these most precisely, however, it turns out to be convenient to generalize from [[2-categories]] to [[F-categories]], using the corresponding notions of $\mathcal{F}$-monad, $\mathcal{F}$-limit, and so on.

## Definition

Let $D$ be a [[small category|small]] strict $\mathcal{F}$-[[F-category|category]].  Then we have the [[functor category|functor]] $\mathcal{F}$-category $[D,\mathcal{F}]$ (where $\mathcal{F}$ denotes the $\mathcal{F}$-category $\mathcal{F}$).  An object of $[D,\mathcal{F}]$ is an $\mathcal{F}$-functor $\Phi\colon D\to \mathcal{F}$, which can be identified with a pair of 2-functors $\Phi_\tau\colon D_\tau \to Cat$ and $\Phi_\lambda\colon D_\lambda\to Cat$ together with a 2-natural transformation
$$\array{D_\tau & & \overset{J_D}{\to} & & D_\lambda\\
  & {}_{\Phi_\tau}\searrow & \neArrow & \swarrow_{\Phi_\lambda} \\
  & & Cat } $$
whose components are full embeddings (objects of $\mathcal{F}$).

The [[tight morphisms]] in $[D,\mathcal{F}]$ are $\mathcal{F}$-natural transformations in the usual sense of enriched category theory, whereas its [[loose morphisms]] are 2-natural transformations between loose parts.

We also have an $\mathcal{F}$-category $Oplax(D,\mathcal{F})$ with the same objects, whose loose morphisms are [[oplax natural transformations]] between loose parts which are strictly 2-natural on tight morphisms, and whose tight morphisms are those whose components are all tight.  The inclusion
$$ [D,\mathcal{F}] \to Oplax(D,\mathcal{F}) $$
has a left adjoint, which induces an $\mathcal{F}$-[[comonad]] $\mathcal{Q}_c^D$ on $[D,\mathcal{F}]$.

+-- {: .un_defn}
###### Definition
A weight $\Phi\colon D\to \mathcal{F}$ is **$l$-rigged** if

1. It is a $\mathcal{Q}_c^D$-coalgebra, and
1. The canonical functor $Lan_{J_D} \Phi_\tau \to \Phi_\lambda$ is surjective on objects.
=--

We obtain definitions of **$c$-rigged** and **$p$-rigged** weights if we replace $Oplax(D,\mathcal{F})$ by $Lax(D,\mathcal{F})$ and $Pseudo(D,\mathcal{F})$, respectively.

## Characterization

Let $w$ denote one of $l$, $c$, or $p$.

+-- {: .un_theorem}
###### Theorem
For an $\mathcal{F}$-weight $\Phi$, the following are equivalent.

1. $\Phi$ is $w$-rigged.
1. For any $\mathcal{F}$-monad $T$ on an $\mathcal{F}$-category $K$, the $\mathcal{F}$-functor $U_w\colon T Alg_w \to K$ creates $\Phi$-weighted limits.
1. For any 2-monad $T$ on a 2-category $K$, the functor $U_w\colon T Alg_w \to K$ (where $K$ denotes the chordate $\mathcal{F}$-category on $K$) creates $\Phi$-weighted limits.
=--

See ([LS](#LS)) for the proof.

## Examples

The following limits are $l$-rigged.

* The [[2-limit]] of any diagram of tight morphisms which is also a limit as a diagram of loose morphisms.  This includes any [[product]] and any [[power]].

* The [[oplax limit]] of any diagram of loose morphisms.

* The [[inserter]] of a [[parallel pair]] $f,g\colon A\to B$ such that $f$ (the domain of the 2-cell to be inserted) is tight.  Here the projection to $A$ is tight and tightness-detecting.

* The [[equifier]] of a parallel pair of 2-cells between a parallel pair of 1-morphisms $f,g\colon A\to B$ such that $f$ (the domain of the 2-cells) is tight.  Again, the projection to $A$ is tight and tightness-detecting.

* The [[Eilenberg-Moore object]] of a loose monad.  Here the canonical forgetful morphism is tight and tightness-detecting.

Each has a fairly obvious dual version which is $c$-rigged.  There are $p$-rigged versions as well, but $p$-rigged weights are almost equivalent to [[PIE-limits]]; see ([LS](#LS)) for details.

## Related concepts

* Some limits of diagrams involving both lax and colax morphisms can also be given a $T$-algebra structure; see for instance [[colax/lax comma object]].

## References

* [[Stephen Lack]], "Limits for lax morphisms". *Appl. Categ. Structures*,
13(3):189--203, 2005

* [[Stephen Lack]] and [[Mike Shulman]], "Enhanced 2-categories and limits for lax morphisms", [arXiv](http://arxiv.org/abs/1104.2111).
 {#LS}

[[!redirects rigged limit]]
[[!redirects rigged limits]]
[[!redirects l-rigged limit]]
[[!redirects l-rigged limits]]
[[!redirects c-rigged limit]]
[[!redirects c-rigged limits]]
[[!redirects p-rigged limit]]
[[!redirects p-rigged limits]]
[[!redirects rigged weight]]
[[!redirects rigged weights]]


right adjoint

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## Idea

The right [[adjoint functor]] of a [[functor]], if it exists, is one of two best approximations to a [[weak inverse]] of that functor.  (The other best approximation is the functor\'s [[left adjoint]], if it exists.)  Note that a weak inverse itself, if it exists, must be a right adjoint, forming an [[adjoint equivalence]].

A right adjoint to a [[forgetful functor]] is called a [[cofree functor]]; in general, right adjoints may be thought of as being defined cofreely, consisting of anything that works in an inverse, regardless of whether it's needed.

The concept generalises immediately to [[enriched category|enriched categories]] and in [[2-category|2-categories]].

## Definitions

Given [[partial order|posets]] (or [[preorder|prosets]]) $C$ and $D$ and a monotone function $U: C \to D$, a __right adjoint__ of $U$ is a monotone function $G: D \to C$ such that
$$ U(x) \leq y \;\Leftrightarrow\; x \leq G(y) $$
for all $x$ in $C$ and $y$ in $D$.

Given [[locally small categories]] $C$ and $D$ and a [[functor]] $U: C \to D$, a __right adjoint__ of $U$ is a functor $G: D \to C$ with a [[natural isomorphism]] between the [[hom-set]] functors
$$ Hom_D(U(-),-), Hom_C(-,G(-)): C^op \times D \to Set .$$

Given $V$-enriched categories $C$ and $D$ and a $V$-[[enriched functor]] $U: C \to D$, a __left adjoint__ of $U$ is a $V$-enriched functor $F: D \to C$ with a $V$-[[enriched natural transformation|enriched natural isomorphism]] between the [[hom-object]] functors
$$ Hom_C(F(-),-), Hom_D(-,U(-)): D^op \times C \to Set .$$

Given categories $C$ and $D$ and a functor $U: C \to D$, a __right adjoint__ of $U$ is a functor $G: D \to C$ with [[natural transformations]]
$$ \iota: id_C \to U ; G,\; \epsilon: G ; U \to id_D $$
satisfying certain [[triangle identities]].

Given a [[2-category]] $\mathcal{B}$, objects $C$ and $D$ of $\mathcal{B}$, and a morphism $U: C \to D$ in $\mathcal{B}$, a __right adjoint__ of $U$ is a morphism $G: D \to C$ with $2$-morphisms
$$ \iota: id_C \to U ; G,\; \epsilon: G ; U \to id_D $$
satisfying the triangle identities.

Although it may not be immediately obvious, these definitions are all compatible.

Whenever $G$ is a right adjoint of $U$, we have that $U$ is a [[left adjoint]] of $G$.

## Properties

* [[right adjoints preserve limits]]

* right adjoints preserve [[monomorphisms]].

## Related concepts

*  See [[Galois connection]] for right adjoints of monotone functions.
*  See [[adjoint functor]] for right adjoints of functors.
*  See [[adjunction]] for right adjoints in $2$-categories.
*  See [[examples of adjoint functors]] for examples.

* [[parametric right adjoint]]

[[!redirects right adjoints]]

[[!redirects right adjoint functor]]
[[!redirects right adjoint functors]]