differential form

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### Context
#### Differential geometry
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[[!include synthetic differential geometry - contents]]
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# Differential forms
* table of contents
{: toc}

## Idea

A differential form is a geometrical object on a [[manifold]] that can be integrated. A differential form $\omega$ is a [[section]] of the [[exterior algebra]] $\Lambda^* T^* X$ of a [[cotangent bundle]], which makes sense in many contexts (e.g. [[manifolds]], [[algebraic varieties]], [[analytic space|analytic spaces]], ...).

What we\'re actually describing here are the *exterior* differential forms; for more general concepts, see [[absolute differential form]] and [[cogerm differential form]].

## Definitions

### Standard definition

Given a differentiable [[manifold]] $X$, or even a [[generalized smooth space]] $X$ for which this definition makes sense, a __differential form__ on $X$ is a [[section]] of the [[exterior algebra]] of the [[cotangent bundle]] over $X$; sometimes one refers to an __exterior differential form__ to be more precise.  One often requires differential forms to be smooth, or at least continuous, but we will state this explicitly when we want it.  A __differential $p$-form__ on $X$ is a section of the $p$th [[exterior power]] of the cotangent bundle; the [[natural number]] $p \geq 0$ is the __rank__ of the form.  A general differential form is a $p$-indexed [[sequence]] of differential $p$-forms of which all but finitely many are zero; on a finite-dimensional manifold, this latter condition is automatic.

The _space_ $C^\infty\Omega^*(X)$ of *smooth* forms on $X$ may also be defined as the [[universal differential envelope]] of the space $C^\infty\Omega^0(X)$ of smooth functions on $X$ (which are the same as the smooth $0$-forms as defined above); more concretely:

It is generated by the smooth functions and three operations:

*  an associative binary operation of addition generalising the usual addition of functions,
*  an associative binary operation $\wedge$ (the __exterior product__ or __[[wedge product]]__) generalising the usual multiplication of functions,
*  a unary operation $\mathrm{d}$ (the __exterior derivative__, or __differential__);

subject to these identities:

*  addition makes $C^\infty\Omega^*(X)$ into an [[abelian monoid]],
*  both $\mathrm{d}$ and $\wedge$ distribute over addition,
*  $\mathrm{d}\mathrm{d}f = 0$,
*  $\mathrm{d}f \wedge \mathrm{d}f = 0$,
*  $\mathrm{d}(f \wedge \eta) = \mathrm{d}f \wedge \eta + f \wedge \mathrm{d}\eta$,
*  $\mathrm{d}(\mathrm{d}f \wedge \eta) + \mathrm{d}f \wedge \mathrm{d}\eta = 0$,

in which $f$ is a smooth function and $\eta$ is an arbitrary smooth form.  (Note that one often drops the '$\wedge$' after a $0$-form; thus, $f \eta = f \wedge \eta$.  There is hardly any ambiguity if one drops the '$\wedge$' entirely, but it\'s traditional.)

Although not directly stated, it can be proved that addition makes $C^\infty\Omega^*(X)$ into an [[abelian group]]; in fact, it is a [[module]] of the [[commutative ring]] of smooth functions on $X$.  This is further a [[graded module]], graded by the [[natural numbers]], with the elements of grade $p$ being the $p$-forms defined earlier; the space of these is $C^\infty\Omega^p(X)$.  If $\omega$ is a $p$-form and $\eta$ is a $q$-form, we have:

*  $\mathrm{d}\omega$ is a $(p+1)$-form,
*  $\omega \wedge \eta$ is a $(p+q)$-form.

The law
$$\mathrm{d}\mathrm{d}\eta = 0$$
holds for any form $\eta$, but the other laws become more complicated; if $\omega$ is a $p$-form and $\eta$ is a $q$-form, then we get

*  $\omega \wedge \omega = 0$ if $p$ is odd,
*  $\omega \wedge \eta = (-1)^{pq}\, \eta \wedge \omega$,
*  $\mathrm{d}(\omega \wedge \eta) = \mathrm{d}\omega \wedge \eta + (-1)^p\, \omega \wedge \mathrm{d}\eta$.

That is, $C^\infty\Omega^*(X)$ is a [[skew-commutative algebra]] over the ring of smooth functions, equipped with a [[derivation]] $\mathrm{d}$ of degree $1$.  In fact, the description above in terms of generators and relations makes it the [[free functor|free]] skew-commutative algebra over that ring equipped with such a derivation.  (Or if it doesn\'t, then it\'s because I left something out of that description.)

More general forms (in $\Omega^*(X)$) can be recovered as sums of terms, each of which is the wedge product of a function and a smooth form.  (This can also be seen as a special case of a vector-valued form as below.)  One can still define the exterior derivative of a $C^1$ (once continuously differentiable) form; in general, the differential of a $C^k$ form is a $C^{k-1}$ form.  If $X$ is not a smooth manifold but only $C^k$ for some $1 \leq k \lt \infty$, then one has to take more care here, but the definition of the skew-commutative algebra of $C^k$ differential forms can still be made to work.

Given local coordinates $(x^1, \ldots, x^n)$ on a patch $U$ in an $n$-dimensional manifold $X$, any differential form $\eta$ on $U$ can be expressed uniquely as a sum of $2^n$ terms
$$ \eta = \sum_I \eta_I \wedge \mathrm{d}x^I ,$$
where $I$ runs over *increasing* [[lists]] of indices from $(1,\ldots,n)$, each $\eta_I$ is a function on $U$ (continuous, smooth, etc according as $\eta$ is), and
$$ \mathrm{d}x^I = \mathrm{d}x^{I_1} \wedge \cdots \wedge \mathrm{d}x^{I_p} $$
(for $p$ the length of the list $I$) is simply an abbreviation.  For a $p$-form, there are $\left(n \atop p\right)$ terms that appear.

### Twisted and vector-valued forms {#twisted}

Recall that a differential form on $X$ is a section of the exterior algebra of the cotangent bundle over $X$; call this bundle $\Lambda$.  Then given any [[vector bundle]] $V$ over $X$, a __$V$-valued form__ on $X$ is a section of the vector bundle $V \otimes \Lambda$.  The wedge product of a $V$-valued form and a $V'$-valued form is a $(V \otimes V')$-valued form, but if there is a commonly used multiplication map $V \otimes V' \to W$, then we may think of their wedge product as a $W$-valued form.

Of particular importance are $L$-valued forms when $L$ is a [[line bundle]]; these are also called __$L$-twisted forms__.  (Compare the notion of [[twisted form]] in a more general context.)  In local coordinates, a twisted form looks just like an ordinary form, once you choose a nonzero vector in $L$ as a basis.  Therefore, they can seem sneaky and confusing sometimes when you realise that they do not behave in the same way!

Let $\Psi$ be the [[pseudoscalar]] bundle, obtained as the bundle [[associated bundle|associated]] to the frame bundle by the character $\rho(A)=sgn(det(A))$. That is, a section of $\Psi$ (a pseudoscalar field) is given locally by a simple [[scalar]] field (a real-valued function) for each [[orientation]] of a local patch, with opposite orientations giving oppositely-signed scalars.  A $\Psi$-twisted form is called __pseudoform__.

On an $n$-dimensional manifold $X$, the bundle $\Lambda^n(X)=\Lambda^n(T^*M)$ is itself a line bundle, which is associated to the frame bundle by the character $A\mapsto det(A)^{-1}$; a $p$-form twisted by this line bundle is called a __densitised form__. Sometimes an $n$-form is itself called a __density__. As we will see under [integration](#integration) below, it is really an $n$-*pseudo*form (a section of the bundle $\Lambda^n\otimes\Psi$, which comes from the character $A\mapsto |\det(A)|^{-1}$) that should be called a density, but that is not the traditional terminology.

Given any real number $w$, the bundle $|\Lambda|^w$ associated to the character $A\mapsto |det A|^{-w}$ is called the line bundle of $w$-weighted scalars (also called [[density|$w$-densities]]); a form twisted by this line bundle is a __$w$-weighted form__. Note that $|\Lambda|^0=1$ is the trivial bundle, so a $0$-weighted form is just an ordinary form.
Also, an $n$-pseudoform turns out to be equivalent to a $1$-weighted scalar:
$$\Lambda^n \otimes \Psi \simeq |\Lambda|^1.$$
Since $\Psi\otimes\Psi=1$ is the trivial bundle, we have $\Lambda^n \simeq |\Lambda|^1 \otimes\Psi$, and thus a densitised form is equivalent to a $1$-weighted pseudoform:
$$\Lambda^k \otimes \Lambda^n \simeq \Lambda^k \otimes \Psi \otimes |\Lambda|^1$$

The bundle $\Lambda^n\otimes\Psi\simeq|\Lambda|^1$ of $n$-pseudoforms is the [[absolute value]] of the bundle of $n$-forms (that is of densitised $0$-forms), so we may take the absolute value of one of either and get one of the latter.  Similarly, the bundle of $0$-forms is the absolute value of the bundle $\Psi$ of pseudo-$0$-forms;
that is, the trivial bundle is the absolute value of the pseudoscalar bundle.

### As smooth functors on $n$-paths

One way to exhibit this statement nicely is:
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A differential $n$-form on $X$ is a smooth $n$-[[n-functor|functor]] $P_n(X) \to \mathbf{B}^n \mathbb{R}$ from the [[path n-groupoid]] of $X$ to the $n$-fold [[delooping]] of the additive [[Lie group]] of real numbers.
=--
+-- {: .query}
Urs, do you know where the need for orientation comes in here?  I don\'t follow it in enough detail to see, although I intend to read Moerdijk--Reyes.  ---Toby

[[Eric]]: I'm probably confused, but if $\sigma_n$ is a morphism in $P_n(X)$, then (unless $X$ is a [[directed space]]), the opposite $\sigma_n^{-1}$ is also in $P_n(X)$ and I think $\omega(\sigma_n) = -\omega(\sigma_n^{-1})$.

_Toby_:  Between Eric\'s comment here and Urs\'s at [[latest changes]], I\'m happy to remove this query box.

[[Eric]]: I think it is a good question. Maybe we should keep the query box here until the answer is incorporated in the page.

[[Urs Schreiber]]: here is my reply, that I originally posted at [[latest changes]]. I'll try to eventually work this into the main text of the entry

The orientation of the diffential form corresponds to the inherent orientation of [[k-morphism|k-morphisms]]: as we identify the differential form with a smooth functor on the [[path n-groupoid]], that [[path n-groupoid]] necessarily has _oriented_ $k$-volumes as its [[k-morphism|k-morphisms]], simply because these $k$-morphisms need to come with information about their (higher categorical) source and target.

To get pseudo-differential forms that may be integrated also over non-oriented and possibly non-orientable manifolds one needs to consider parallel transport functors not with coefficients in just $\mathbf{B}^n \mathbb{R}$ coming from the [[crossed complex]]

$$
   (\mathbb{R} \to {*} \to \cdots \to {*})
$$

but the more refined crossed complex

$$
  (\mathbb{R} \to {*} \to \cdots \to \mathbb{Z}_2)
$$

where the $\mathbb{Z}_2$-factor acts by sign reversal on $\mathbb{R}$ (one can also use $U(1)$ instead of $\mathbb{R}$, this way $[P_n(-), \mathbf{B}^n U(1)]$ becomes the [[Deligne cohomology|Deligne complex]] and knows not just about differential forms but about $U(1)$ $n-1$-gerbes with connection even).

A little bit of discussion of this unoriented case is currently at [[orientifold]]. There for the case $n=2$.
=--

Note that an $n$-morphism in $P_n(X)$ is an oriented $n$-dimensional submanifold of $X$.

Such a functor (as described in more detail at [[connection on a bundle]]) assigns a real number to each parametrised $n$-dimensional cube of $X$, that is a subspace by a smooth map $\Sigma\colon [0,1]^n \to X$.  If the differential form that this $n$-functor defines is denoted $\omega \in \Omega^n(X)$, then this real number is denoted by the [[integration|integral]]
$$
  \int_{[0,1]^n} \Sigma^* \omega
  \,.
$$

This integral in turn encodes the $n$-functoriality of the $n$-functor: it effectively says that

* if we decompose the standard $n$-cube $[0,1]^n$ into $N^n$ little subcubes $(C_k)_{k\in \mathbb{N}^n}$  for $N \in \mathbb{N}$
* and apply the $n$-functor to each of these to obtain a result (a real number) to be denoted $\omega(C_k)$;
* then by $n$-functoriality the result of the application of the functor to the full $\Sigma$ is the composition of all the $\omega(C_k)$ in $\mathbb{R}$. i.e. their sum
  $$
    \int_{[0,1]^n} \Sigma^* \omega
    =
    \sum_k \omega(C_k)
    \,.
  $$

Since one can let $N$ increase arbitrarily in this prescription -- $N \to \infty$ -- it follows that the value of the functor on $\Sigma$ is already determined by all its values on all "infinitesimal $n$-cubes" in some sense.

The notion of **differential form** is the one that makes this precise: a differential form is a rule for assigning to each "infinitesimal $n$-cube" a number.

There are in turn different ways to make that last statement precise:

* In [[differential geometry]] an "infinitesimal $n$-cube" is modeled by an $n$-tuple of [[tangent bundle|tangent vectors]] and a differential form is a fiberwise linear map from the $n$-fold exterior power of the [[tangent bundle]] to the real numbers, as given below.

* In [[synthetic differential geometry]] the statement is in essence the same one, but the difference is that there the notion of "infinitesimal $n$-cube" has a concrete meaning on the same footing of other $n$-cubes. If $D^n$ denotes the abstract infinitesimal $n$-cube in this context, then the mapping space $X^{D^n}$ of morphisms from $D^n\to X$ is the $n$-fold [[tangent bundle]] of $X$ and a differential form is precisely nothing but a morphism
  $$
    \omega\colon X^{D^n} \times D^n \to \mathbb{R}
  $$
  (where $\mathbb{R}$ is now the synthetic differential version of the real numbers) subject to three constraints. (These constraints can be seen as the infinitesimal analog of the $n$-functoriality discussed above).

  This is described in detail in section 4 of

  * Moerdijk-Reyes, [[Models for Smooth Infinitesimal Analysis]]

  For more on this see [[differential forms in synthetic differential geometry]].

### More general kinds of forms

While differential forms are usually restricted to the linear case, there are also more general kinds of "differential forms" which can be integrated.  See for instance [[absolute differential form]] and [[cogerm differential form]].

## Operations on differential forms

### Pulling back forms

Given manifolds $X$ and $Y$ and a [[continuously differentiable map]] $f\colon X \to Y$, any differential form $\eta$ on $Y$ defines a __pullback__ form $f^*(\eta)$ on $X$.  See at _[[pullback of differential forms]]_.

Thus, the operation that maps $X$ to $\Omega^*(X)$ extends to a [[contravariant functor]] $\Omega^*$.  Perhaps confusingly, forms are traditionally known in physics as 'covariant' concepts, because of how the components transform under a change of coordinates.  (Ultimately, this confusion goes back to that between active and passive [[coordinate transformation|coordinate transformations]].)

Note that twisted and (more general) vector-valued forms cannot be pulled back so easily.  One needs some extra structure on $f$ to do so; see the discussion of integration of $p$-pseudoforms at [[integration of differential forms]] for an example.

### Integration of forms {#integration}

Let $X$ be an $n$-dimensional manifold, and let $\omega$ be an $n$-pseudoform on $X$.  At least when $X$ is [[paracompact space|paracompact]] and [[Hausdorff space|Hausdorff]], we may turn $\omega$ into a [[measure]] on $X$ and thereby find its [[integral]].  Conversely, any [[absolutely continuous measure|absolutely continuous]] [[Radon measure]] on $X$ arises in this way from a unique $n$-pseudoform $\omega$.

If we wish to integrate untwisted (or differently twisted or vector-valued) forms and/or forms of smaller rank, then we may do so on [[submanifolds]] of $X$ equipped with some appropriate structure.  In particular, if $X$ itself is equipped with an [[orientation]], then $n$-pseudoforms on $X$ are the same as (untwisted) $n$-forms, and so we can integrate those on $X$.  See [[integration of differential forms]] for the general case.

+-- {: .query}
[[Zoran Škoda]]: Should maybe this entry have a discussion on heuristics behind the usual trick in supersymmetry which asserts that the inner hom for supermanifolds, gives the statement that the algebra of smooth differential forms on $M$ is the space of functions on the odd tangent bundle $\Pi T M$? I am not the most competent to do this succinctly enough...

_Toby_:  Possibly that should go at [[differential forms on supermanifolds]]?

[[Zoran Škoda]]: By no means. Ordinary differential forms on ORDINARY manifolds are the same as functions on odd tangent bundle. I did not want to say anything about the generalization of differential forms on  supermanifolds. So it is NOT a different notion, but a different way to define it. If going to toposes hence synthetic framework is not separated why would be separated the equality which involves a parity trick...

_Toby_:  Ah, I see; your $M$ above need not be super, and it still works.  Then yes, that should be mentioned here too.
=--

### Cohomology of forms

There is a [[cohomology]] theory of smooth differential forms; we have a [[chain complex]]
$$ \cdots \stackrel{\mathrm{d}}\to C^\infty\Omega^2(X) \stackrel{\mathrm{d}}\to C^\infty\Omega^1(X) \stackrel{\mathrm{d}}\to C^\infty\Omega^0(X) \stackrel{\mathrm{d}}\to 0 ;$$
the [[chain cohomology]] of this complex is the __[[de Rham cohomology]]__ of $X$.

As smooth differential forms are the cochains in de Rham cohomolgy, the theory of integration of forms allows us to interpret [[relatively compact subspace|relatively compact]] oriented submanifolds as chains on $X$, giving us a [[homology]] theory.  Combining these, we have __[[Stokes's theorem]]__
$$ \int_{\partial{R}} \omega = \int_R \mathrm{d}\omega ,$$
where $\partial{R}$, which may be interpreted as the __boundary__ of $R$, is also called the __codifferential__ as it is dual to $\mathrm{d}$.

## Related concepts

* [[closed differential form]], [[exact differential form]]

* [[wedge product]]

* [[decomposable differential form]]

* [[integration of differential forms]]

* [[horizontal differential form]], [[vertical differential form]]

* [[harmonic differential form]]

* [[differential form on a supermanifold]]

* [[Kähler differential form]]

* [[de Rham complex]], [[de Rham cohomology]]

* [[differential cohomology]]

* [[absolute differential form]], [[cogerm differential form]]

* [[holomorphic differential form]]

* [[differential form with logarithmic singularities]]

## References

### General

The concept of differential forms (and their [[exterior algebra]]), at least on [[affine spaces]]/[[Euclidean spaces]] originates in

* [[Hermann Grassmann]], _[[Ausdehnungslehre]]_, 1844

Textbook accounts:

* [[Raoul Bott]], [[Loring Tu]], *[[Differential Forms in Algebraic Topology]]*, Graduate Texts in Mathematics 82, Springer (1982) [[doi:10.1007/978-1-4757-3951-0](https://link.springer.com/book/10.1007/978-1-4757-3951-0)]

* [[Georges de Rham]], Chapter II of: *Differentiable Manifolds -- Forms, Currents, Harmonic Forms*, Grundlehren **266**, Springer (1984) [[doi:10.1007/978-3-642-61752-2](https://doi.org/10.1007/978-3-642-61752-2)]

* [[Manfredo P. Carmo]], *Differential Forms and Applications*, Springer (1994) [[doi:10.1007/978-3-642-57951-6](https://doi.org/10.1007/978-3-642-57951-6)]

* {#GuilleminHaine19} [[Victor Guillemin]], [[Peter Haine]], _Differential Forms_, World Scientific (2019) [[doi:10.1142/11058](https://doi.org/10.1142/11058), draft:[pdf](https://math.mit.edu/classes/18.952/2018SP/files/18.952_book.pdf)]

A basic introduction with an eye towards applications in [[physics]]:

* [[Theodore Frankel]], section 2.1 of: *[[The Geometry of Physics - An Introduction]]* (1997)

An introductory wiki-format textbook is

* {#Dray_GDF} Tevian Dray, _The Geometry of Differential Forms_, [web](http://physics.oregonstate.edu/coursewikis/GDF/)

published as half of

* Tevian Dray, _Differential Forms and the Geometry of General Relativity_, A K Peters/CRC Press, 2014,
 [web](http://physics.oregonstate.edu/coursewikis/DFGGR/bookinfo/).

The equivalence between differential forms and smooth functors on the [[path groupoid]] in low degree is discussed in

* Schreiber & Waldorf, _Smooth Functors vs. Differential Forms_ ([arXiv](http://arxiv.org/abs/0802.0663))

Much fun discussion between [[Eric Forgy]], [[Toby Bartels]], and [[John Baez]], about whether integration of forms or pseudoforms is most fundamental (and about whether twisted forms in general are useful and interesting geometric objects or the bastard spawn of hell) may be found in [this giant Usenet thread](https://groups.google.com/group/sci.physics.research/browse_thread/thread/6a231426b3a313c0/2888a120a9b1f5ad).  More specifically:

*  [John\'s alter ego the Wizard](https://groups.google.com/group/sci.physics.research/msg/3c6a1a7237b66c8c?dmode=source) explains why $n$-pseudoforms are the most natural things to integrate on an $n$-dimensional manifold;
*  [applications of pseudoforms](https://groups.google.com/group/sci.physics.research/msg/2774cbbc982e200e?dmode=source) to classical contexts where absolute values appear;
*  [the general notion of form](https://groups.google.com/group/sci.physics.research/msg/47bbd29289f208f8?dmode=source) corresponding to an arbitrary representation of the [[general linear group]] (towards the end of the post);
*  [absolute values of forms](https://groups.google.com/group/sci.physics.research/msg/424da828e75b6b90?dmode=source) and how to integrate them (near the end of the post, a precursor to [[absolute differential forms]]).

[[!include electromagnetism in terms of differential forms -- references]]

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differential form on a supermanifold

+-- {: .rightHandSide}
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###Context###
#### Super-Geometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The notion of superdifferential form is the generalization of the notion of [[differential form]] from [[manifolds]] to [[supermanifolds]].

## Definition
 {#Definition}

+-- {: .num_defn #DifferentialFormsOnSuperCartesianSpace}
###### Definition
**([[differential forms]] on [[super Cartesian space]])**

For $n,q \in \mathbb{N}$, the _de Rham algebra_ $\Omega^\bullet(\mathbb{R}^{n\vert q})$ of super-differential forms on
the [[super Cartesian space]] $\mathbb{R}^{n\vert q}$ is the [[free construction|free]] [[differential graded-commutative superalgebra]] over the [[supercommutative algebra]]

$$
  C^\infty(\mathbb{R}^{n\vert q})
  \;=\;
  \underset{
    even
  }{
  \underbrace{
    C^\infty(\mathbb{R}^n)
  }} \otimes_{\mathbb{R}}
  \underset{
    odd
  }{
  \underbrace{
    \wedge^\bullet \mathbb{R}^q
  }}
$$

on

1. $n$ [[generators and relations|generators]] $\mathbf{d} x^a$ in [bi-degree](chain+complex+in+super+vector+spaces#ChainComplexesOfSuperVectorSpaces) $(1,even)$ (the canonical bosonic 1-forms)

1. $q$ generators $\mathbf{d} \theta^\alpha$ in [bi-degree](chain+complex+in+super+vector+spaces#ChainComplexesOfSuperVectorSpaces) $(1,odd)$

hence:

\[
  \label{SuperCartSpDiffForms}
  \Omega^\bullet( \mathbb{R}^{n\vert q} )
  \;\coloneqq\;
  C^\infty(\mathbb{R}^{n\vert q})
  \left[
    \underset{deg = (1,even)}{\underbrace{
    \left\langle
      \mathbf{d} x^a
    \right\rangle_{a = 1}^{n}
    }},\;\;\;\;\;
    \underset{
      deg = (1,odd)
    }{
    \underbrace{
    \left\langle
      \mathbf{d} \theta^\alpha
    \right\rangle_{\alpha = 1}^q
    }}
  \right]
\]

with [[nLab:differential]] having the evident definition on generators, and extended from there as a [[derivation]] of bi-degree $(1, even) \in \mathbb{Z} \times (\mathbb{Z}/2)$

=--

+-- {: .num_example #SignRulesForDifferentialFormsOnSuperCartesianSpaces}
###### Example
**([[signs in supergeometry|sign rule]] for [[differential forms]] on [[super Cartesian spaces]])**

For $n,q \in \mathbb{N}$, the generators of the [[differential graded-commutative superalgebra]] $\Omega^\bullet(\mathbb{R}^{n\vert q})$ of differential forms on the [[super Cartesian space]] (Def. \ref{DifferentialFormsOnSuperCartesianSpace}) have the following [bi-degree](chain+complex+in+super+vector+spaces#ChainComplexesOfSuperVectorSpaces)

| $\phantom{A}$generator$\phantom{A}$ | $\phantom{A}$[bi-degree](chain+complex+in+super+vector+spaces#ChainComplexesOfSuperVectorSpaces)$\phantom{A}$ |
|--|--|
| $\phantom{A}$$x^a$$\phantom{A}$ | $\phantom{A}$(0,even)$\phantom{A}$ |
| $\phantom{A}$$\theta^\alpha$$\phantom{A}$ | $\phantom{A}$(0,odd)$\phantom{A}$ |
| $\phantom{A}$$\mathbf{d}x^a$$\phantom{A}$ | $\phantom{A}$(1,even)$\phantom{A}$ |
| $\phantom{A}$$\mathbf{d}\theta^\alpha$$\phantom{A}$ | $\phantom{A}$(1,odd)$\phantom{A}$ |
{: style='margin:auto'}

and satisfy the following graded-commutation relations, depending on one of the two equivalent (see [here](chain+complex+in+super+vector+spaces#EquivalenceTwoSymmetricMonoidalStructuresOnChSuperVect)) [[signs in supergeometry|sign rules]]:

| $\phantom{A}$[[signs in supergeometry|sign rule]]$\phantom{A}$  | $\phantom{A}$[Deligne's](signs+in+supergeometry#TheSignRuleFromInternalization)$\phantom{A}$ | [Bernstein's](signs+in+supergeometry#SuperOddConvention)$\phantom{A}$ |
|---|------------------|-------------------|
| $\phantom{A}$$x^{a} \; x^{b} =$ | $\phantom{A}$$+ x^{b} \; x^{a}$$\phantom{A}$ | $\phantom{A}$$+ x^{b} \; x^{a}$$\phantom{A}$ |
| $\phantom{A}$$x^a \;\theta^\alpha =$$\phantom{A}$ | $\phantom{A}$$+ \theta^\alpha \; x^a$$\phantom{A}$ | $\phantom{A}$$+ \theta^\alpha \; x^a$$\phantom{A}$ |
| $\phantom{A}$$\theta^{\alpha} \; \theta^{\beta} =$$\phantom{A}$ | $\phantom{A}$$- \theta^{\beta} \; \theta^{\alpha}$$\phantom{A}$ | $\phantom{A}$$ - \theta^{\beta} \; \theta^{\alpha}$$\phantom{A}$ |
| $\phantom{A}$$x^{a} (\mathbf{d}x^{a}) =$$\phantom{A}$ | $\phantom{A}$$+ (\mathbf{d}x^{b}) x^{a}$$\phantom{A}$ | $\phantom{A}$$+ (\mathbf{d}x^{b}) x^{a}$$\phantom{A}$ |
| $\phantom{A}$$\theta^\alpha (\mathbf{d}x^a) =$$\phantom{A}$ | $\phantom{A}$$+ (\mathbf{d}x^a) \theta^\alpha$$\phantom{A}$ | $\phantom{A}$${\color{blue}{-}} (\mathbf{d}x^a) \theta^\alpha$$\phantom{A}$ |
| $\phantom{A}$$\theta^{\alpha} (\mathbf{d}\theta^{\beta})  = $$\phantom{A}$ | $\phantom{A}$$- (\mathbf{d}\theta^{\beta}) \theta^{\alpha}$$\phantom{A}$ | $\phantom{A}$${\color{blue}{+}} (\mathbf{d}\theta^{\beta}) \theta^{\alpha}$$\phantom{A}$ |
| $\phantom{A}$$ (\mathbf{d}x^{a}) (\mathbf{d} x^{b}) =$$\phantom{A}$ |  $\phantom{A}$$- (\mathbf{d} x^{b}) (\mathbf{d} x^{a})$$\phantom{A}$ | $\phantom{A}$$ - (\mathbf{d} x^{b}) (\mathbf{d} x^{a})$$\phantom{A}$ |
| $\phantom{A}$$ (\mathbf{d}x^a) (\mathbf{d} \theta^{\alpha}) =$$\phantom{A}$ | $\phantom{A}$$ - (\mathbf{d}\theta^{\alpha}) (\mathbf{d} x^a) $$\phantom{A}$ | $\phantom{A}$$ {\color{blue}{+}} (\mathbf{d}\theta^{\alpha}) (\mathbf{d} x^a) $$\phantom{A}$ |
| $\phantom{A}$$(\mathbf{d}\theta^{\alpha}) (\mathbf{d} \theta^{\beta}) =$ | $\phantom{A}$$ + (\mathbf{d}\theta^{\beta}) (\mathbf{d} \theta^{\alpha})$$\phantom{A}$ | $\phantom{A}$$ + (\mathbf{d}\theta^{\beta}) (\mathbf{d} \theta^{\alpha})$$\phantom{A}$ |
{: style='margin:auto'}

=--

+-- {: .num_defn #SuperCartSpaPullbackOfSuperDifferentialForms}
###### Definition
**([[pullback of differential forms|pullback]] over [[super Cartesian spaces]])**

Let

$$
  \array{
    \mathbb{R}^{n_1 \vert q_1}
    &\overset{f}{\longrightarrow}&
    \mathbb{R}^{n_2 \vert q_2}
    \\
    C^\infty(\mathbb{R}^{n_1\vert q_1})
    &\overset{f^\ast}{\longleftarrow}&
    C^\infty(\mathbb{R}^{n_2\vert q_2})
  }
$$

be a morphism of [[super Cartesian spaces]], hence [[formal duality|formally dually]] a [[algebra homomorphism]] $f^\ast$ of [[supercommutative superalgebras]].

By the fact that $\Omega^\bullet(\mathbb{R}^{n \vert q})$ (Def. \ref{DifferentialFormsOnSuperCartesianSpace}) is [[free construction|free]] over $C^\infty(\mathbb{R}^{n\vert q})$ on generators $\mathbf{d}x^a$, $\mathbf{d}\theta^\alpha$ (eq:SuperCartSpDiffForms) this extends to a unique homomorphism on the de Rham [[dgc-superalgebra]]

$$
  \Omega^\bullet(\mathbb{R}^{n_1\vert q_1})
    \overset{f^\ast}{\longleftarrow}
  \Omega^\bullet(\mathbb{R}^{n_2\vert q_2})
$$

subject to the condition that $f^\ast \circ \mathbf{d}_2 = \mathbf{d}_1 \circ f^\ast$:

$$
  \array{
    \Omega^\bullet(\mathbb{R}^{n_1\vert q_1})
    &\overset{\phantom{AA}f^\ast\phantom{AA}}{\longleftarrow}&
    \Omega^\bullet(\mathbb{R}^{n_2\vert q_2})
    \\
    {}^{\mathbf{d}_1}\Big\downarrow
    &&
    \Big\downarrow{}^{\mathbf{d}_2}
    \\
    \Omega^\bullet(\mathbb{R}^{n_1\vert q_1})
    &\overset{\phantom{AA}f^\ast\phantom{AA}}{\longleftarrow}&
    \Omega^\bullet(\mathbb{R}^{n_2\vert q_2})
  }
$$

This operation is called _[[pullback of differential forms]]_ along maps of [[super Cartesian spaces]].

=--

+-- {: .num_prop #SuperDifferentialFormClassifying}
###### Proposition
**(classifing [[super formal smooth set]] of [[super differential forms]])**

The operation of [[pullback of differential forms]] (Def. \ref{SuperCartSpaPullbackOfSuperDifferentialForms}) over [[super Cartesian spaces]] respects [[identity morphisms]] and [[composition]]. Hence the assignment of [[differential forms]] on [[super Cartesian spaces]] (Def. \ref{DifferentialFormsOnSuperCartesianSpace}) is a [[presheaf]] on [[SuperCartSp]]:

$$
  \mathbf{\Omega}\bullet
  \;\colon\;
  SuperCartSp^{op}
    \longrightarrow
  dgcsAlg
$$

with values in [[differential graded-commutative superalgebras]], in fact a [[sheaf]] and hence a [[differential graded algebra]] [[internalization|internal]] to the [[sheaf topos]] over [[SuperCartSp]]:

$$
  \mathbf{\Omega}^n
  \;\in\;
  Sh(SuperCartSp)
  \,.
$$

The construction generalizes in an evident way also to sheaves over [[super formal Cartesian spaces]], hence to [[super formal smooth sets]]:

$$
  \mathbf{\Omega}^n
  \;\in\;
  SuperFormalSmoothSet \;\coloneqq\; Sh(SuperFormalCartSp)
  \,.
$$

=--

We may now proceed as in the discussion of [[differential forms]] on [[smooth sets]] ([here](geometry+of+physics+--+smooth sets#DifferentialForms)):

+-- {: .num_defn #SuperDifferentialFormsOnGeneralSuperFormalSmoothSets}
###### Definition
**([[super differential forms]] on general [[super formal smooth sets]])**

Let $X$ be a [[supermanifold]] or more generally a [[super formal smooth set]]

$$
  X \;\in\; SuperFormalSmoothSet
$$

Then _[[super differential forms]]_ on $X$ are morphisms

$$
  X \longrightarrow \mathbf{\Omega}^\bullet
$$

into the classifying sheaf $\mathbf{\Omega}^\bullet$ from Def. \ref{SuperDifferentialFormClassifying}.

=--

## Properties

### Integral top-forms and Picture number

If a choice of _integral top-forms_ is made, needed for a notion of _[[integration over supermanifolds]]_, then there is an additional grading by "[[picture number]]" ([Belopolsky 97b](#Belopolsky97b), [Witten 12](#Witten12)), see ([Catenacci-Grassi-Noja 18 (5.8) to (5.12)](#CatenacciGrassiNoja18)).

## Examples {#Examples}

Let $\mathbf{X} = \mathbb{R}^{1|1}$. The [[superalgebra]] of functions on $\mathbf{X}$ is the [[exterior algebra]] that is generated over $C^\infty(\mathbf{R})$ from a single generator $\theta$ in odd degree (the canonical odd coordinate).

The algebra of superdifferential forms on $\mathbb{R}^{1|1}$ is the [[exterior algebra]] generated over $C^\infty(\mathbb{R})$ from

* a generator $\theta$ in odd degree (the canonical odd coordinate);

* a generator $d x$ in odd degree (the differential of the canonical even coordinate);

* a generator $d \theta$ in even degree (the differential of the canonical odd coordinate).

Notice in particular that while $ d x \wedge d x = 0$
the wedge product $d \theta \wedge d\theta$ is non-vanishing, since $d \theta$ is in even degree. In fact all higher wedge powers of $d \theta$ with itself exist.

## Remarks

* Being a $\mathbb{Z}_2$-graded locally free algebra itself, one can regard $\Omega^\bullet(X)$ itself (even for $X$ a usual manifold!) as the "algebra of functions" (more precisely [[inner hom]], i.e. mapping space into the line) on another supermanifold. That supermanifold is called $T[1] X$, the **[[shifted tangent bundle]]** of $X$. By definition we have $C^\infty(T[1]X) = \Omega^\bullet(X)$. From this point of view, the existence of the differential $d$ on the graded algebra $\Omega^\bullet(X)$ translates into the existence of a special odd vector field on $T[1]X$. This is a **homological vector field** in that it is odd and the super Lie bracket of it with itself vanishes: $[d,d] = 0$.

* In the context of [[L-infinity algebroids]], where one may regard $C^\infty(X)$ as the [[Chevalley-Eilenberg algebra]] of an $L_\infty$-[[Lie infinity-algebroid|algebroid]] it is useful to notice that $\Omega^\bullet(X)$ is the corresponding [[Weil algebra]]. If $X$ is a Lie $n$-algebroid then $T[1]X$ is a Lie $(n+1)$-algebroid.

## Related concepts

* [[signs in supergeometry]]

* [[picture changing operator]]

* [[super Lie algebroid]]

## References

General discussion:

* [[Simone Noja]], *On the Geometry of Forms on Supermanifolds* ([arXiv:2111.12841](https://arxiv.org/abs/2111.12841))

Discussion in view of [[Lie algebra cohomology]] of [[super Lie algebras]]:

* [[Roberto Catenacci]], [[C. A. Cremonini]], [[Pietro Antonio Grassi]], [[Simone Noja]], _Cohomology of Lie Superalgebras: Forms, Integral Forms and Coset Superspaces_ ([arXiv:2012.05246](https://arxiv.org/abs/2012.05246))

Discussion with an eye towards [[quantization]] of the [[superstring]] is in

* [[Alexander Belopolsky]], _De Rham Cohomology of the Supermanifolds and Superstring BRST Cohomology_, Phys.Lett. B403 (1997) 47-50 ([arXiv:hep-th/9609220](http://arxiv.org/abs/hep-th/9609220))

Geometric discussion of [[picture number]] appearing in the context of [[integration over supermanifolds]] (and originally seen in the [[quantization]] of the [[NSR superstring]], crucial in [[superstring field theory]]) is due to

* {#Belopolsky97b} [[Alexander Belopolsky]], _Picture changing operators in supergeometry and superstring theory_ ([arXiv:hep-th/9706033](https://arxiv.org/abs/hep-th/9706033))

and further amplified in

* {#Witten12} [[Edward Witten]], appendix D of _Notes On Super Riemann Surfaces And Their Moduli_ ([arXiv:1209.2459](http://arxiv.org/abs/1209.2459))

* {#CatenacciGrassiNoja18} [[Roberto Catenacci]], [[Pietro Antonio Grassi]], [[Simone Noja]], _Superstring Field Theory, Superforms and Supergeometry_ ([arXiv:1807.09563](https://arxiv.org/abs/1807.09563))

[[!redirects differential forms on a supermanifold]]

[[!redirects differential form on supermanifold]]
[[!redirects differential forms on supermanifolds]]

[[!redirects super differential form]]
[[!redirects super differential forms]]

[[!redirects superdifferential form]]
[[!redirects superdifferential forms]]


#Contents#
* table of contents
{:toc}

## Idea

A _differential form with logarithmic singularities_ is a [[meromorphic function|meromorphic]] [[differential form]] on some space $X$ which is a [[holomorphic differential form]] on a suitably dense open subspace with at most [[logarithm|logarithmic]] [[singularities]] at the boundary.

These are the differential forms on spaces in [[logarithmic geometry]]. They form the logarithmic generalization of the [[holomorphic de Rham complex]].

## Examples

* The [[moduli spaces of connections]] with logarithmic singularity at prescribed points are for instance the object of interest in "ramified" [[geometric Langlands duality]]. (e.g. [Witten 08, section 4](#Witten08))

* A [['t Hooft operator]] is a [[gauge field]] configuration with certain logarithmic differential form singularities.

## Related concepts

* [[logarithmic geometry]]

* [[logarithmic connection]]

* [[differential algebraic K-theory]]

## References

The brief idea is well described in

* {#Pottharst} Pottharst, _Logarithmic structures on schemes_ ([pdf](http://math.bu.edu/people/potthars/writings/log.str.pdf))

Further details are discussed in

* {#Ogus01} [[Arthur Ogus]], Chapter IV of _Lectures on logarithmic algebraic geometry_, TeXed notes, 2001, [pdf](http://math.berkeley.edu/~ogus/preprints/log_book/logbook.pdf)

* [[Arthur Ogus]], slides 31 ff in _Logarithmic geometry_, talk slides 2009 ([pdf](http://math.berkeley.edu/~ogus/preprints/colloqhandout.pdf))

* {#Burgos} [[José Ignacio Burgos]], _A $C^\infty$-logarithmic Dolbeault complex_, Compositio Math. 92 (1994), no. 1, 61-86. MR 1275721 (95g:32056)

* {#Voisin02} [[Claire Voisin]], section 8.2.2 of _[[Hodge theory and Complex algebraic geometry]] I,II_, Cambridge Stud. in Adv. Math. __76, 77__, 2002/3

Discussion in the context of [[geometric Langlands duality]] includes

* {#Witten08} [[Edward Witten]], _Mirror Symmetry, Hitchin's Equations, And Langlands Duality_ ([arXiv:0802.0999](http://arxiv.org/abs/0802.0999))

In the context of [[differential algebraic K-theory]]

* {#BunkeTamme12} [[Ulrich Bunke]], [[Georg Tamme]], section 3.1 of _Regulators and cycle maps in higher-dimensional differential algebraic K-theory_ ([arXiv:1209.6451](http://arxiv.org/abs/1209.6451))

[[!redirects differential forms with logarithmic singularities]]


This entry collects material related to the book

* [[Raoul Bott]], [[Loring Tu]]:

  \linebreak

  **Differential Forms in Algebraic Topology**

  \linebreak

  Graduate Texts in Mathematics **82**

  Springer (1982)

  [doi:10.1007/978-1-4757-3951-0](https://link.springer.com/book/10.1007/978-1-4757-3951-0)

on applications of [[differential forms]], or rather of [[de Rham cohomology]], in [[algebraic topology]].

Related references:

* {#GuilleminHaine19} [[Victor Guillemin]], [[Peter Haine]], _Differential Forms_, World Scientific (2019) ([doi:10.1142/11058](https://doi.org/10.1142/11058))

#Contents#
* table of contents
{:toc}

## 1 The de Rham complex on $\mathbb{R}^n$

* [[differential forms]]

* [[wedge product]]

* [[exterior differential]]

* [[de Rham complex]]

## 2 The Mayer-Vietoris sequence

## 3 Orientation and integration

* [[volume form]]

* [[integration of differential forms]]

* [[Stokes theorem]]

## 4 Poincaré lemma

* [[Poincare lemma]]

* [[Gysin sequence]]

## 5 The Mayer-Vietoris argument

## 6 The Thom isomorphism

* [[vector bundles]]

* [[fiber integration]], [[projection formula]]

* [[Thom isomorphism]]

## 7 The Nonorientable case

## 8 The Generalized Mayer-Vietoris Principle

## 9 More Examples and Applications of the Mayer-Vietoris Principle

## 10 Presheaves and Cech cohomology

* [[presheaves]]

* [[Cech cohomology]]

## 11 Sphere bundles

* [[spherical fibration]]

* [[Euler form]]

* [[Euler characteristic]]

* [[Poincaré–Hopf theorem]]

* [[Hopf degree theorem]]

## 13 Monodromy

* [[monodromy]]

## 14 The spectral sequence of a filtered complex

* [[spectral sequence]]

* [[spectral sequence of a filtered complex]]

## 17 Review of homotopy theory

* [[homotopy theory]]

* [[homotopy groups of spheres]]

* [[cell attachment]]

* [[Morse theory]]

* [[Hopf invariant]]

## 19 Rational homotopy theory

* [[rational homotopy theory]]

## 20 Chern Classes of a Complex Vector Bundle

* [[complex vector bundle]]

* [[Chern class]]

## 21 The Splitting Principle and Flag manifolds

* [[splitting principle]]

## 22 Pontrjagin classes

* [[Pontrjagin classes]]

## 23 The Search for the Universal Principal Bundle

* [[classifying space]]

* [[universal principal bundle]]

category: reference


differential forms in synthetic differential geometry

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Differential geometry
+-- {: .hide}
[[!include synthetic differential geometry - contents]]
=--
=--
=--

#Content#
* automatic table of contents goes here
{:toc}

#Idea#

In the context of [[synthetic differential geometry]] a [[differential form]] $\omega$ of degree $k$ on a [[manifold]] $X$ is literally a function on the space of _infinitesimal $k$-cubes_ or _infinitesimal $k$-simplices_ in $X$.

We give the definition as available in the literature and then interpret this in a more unified way in terms of the [[schreiber:Chevalley-Eilenberg algebra|Chevalley-Eilenberg algebra]] of the [[infinitesimal singular simplicial complex]].

# Definition #

> missing here are details on what axioms the space we are working on has to satisfy for the following to make sense. See the case distinction at [[infinitesimal singular simplicial complex]].

## differential forms ##

An **infinitesimal $k$-simplex** in a synthetic differential space $X$ is a collection of $k+1$-points in $X$ that are pairwise infinitesimal neighbours.

The spaces $X^{\Delta^k_{diff}}$ of infinitesimal $k$-simplices arrange to form the [[infinitesimal singular simplicial complex]] $X^{\Delta^\bullet_{diff}}$.

The functions on the space of infinitesimal $k$-simplices form a [[generalized smooth algebra]] $C^\infty(X^{\Delta^k_{inf}})$.

A **differential $k$-form** (often called simplicial $k$-form or, less accurately, **combinatorial $k$-form** to distinguish it from similar but cubical definitions) on $X$ is an element in this function algebra that has the property that it vanishes on degenerate infinitesimal simplices.

See definition 3.1.1 in

* Anders Kock, _Synthetic geometry of manifolds_ ([pdf](http://tildeweb.au.dk/au76680/SGM-final.pdf))

for this simplicial definition. A detailed account of this is in the entry [[infinitesimal object]] in the section [Spaces of infinitesimal simplices](http://ncatlab.org/nlab/show/infinitesimal+object#SpacOfInfSimpl).

This is a very simple-looking statement. The reason is the [[topos]]-theoretic language at work in the background, which takes care that we may talk about [[infinitesimal object]]s as if they were just plain ordinary sets. For a very detailed account of how the above statement is implemented concretely in terms of concrete models for synthetic differential spaces see section 1 of

* Breen, Messing, _Combinatorial differential forms_ ([arXiv](http://arxiv.org/abs/math/0005087))

There are also cubical variants of the above definition

* Anders Kock, _Cubical version of combinatorial differential forms_ ([pdf for fee](http://www.springerlink.com/content/87tj80l51h138177/fulltext.pdf))

See also section 4.1 of

* Moerdijk-Reyes,  [[Models for Smooth Infinitesimal Analysis]]

for a realization of the cubical version in models based on sheaves on [[generalized smooth algebra]]s.

We may characterize the object $\Omega^k(X) \subset C^\infty(X^{\Delta^k_{inf}})$ as follows:

for $k \geq 1$ there are the obvious images

$$
  s_i^*
  :
  C^\infty(X^{\Delta^{k}_{inf}})
  \to
  C^\infty(X^{\Delta^{k-1}_{inf}})
$$

of the [[simplicial set|degeneracy maps]]. As one can see, these act by restricting a function on infinitesimal $k$-simplices to the degenerate ones and regarding these then as a $(k-1)$-simplex.

Therefore we may characterize the subobject $\Omega^k(X) \hookrightarrow C^\infty(X^{\Delta^k_{inf}})$ as the joint kernel of the degeneracy maps
$$
  \Omega^k(X) = \cap_{i = 0}^{k-1} ker(s_i^*)
  \,.
$$

## coboundary operator ##

According to section 3.2 of Andres Kock's book, the coboundary operator $d : \Omega^k(X) \to \Omega^{k+1}(X)$ sends a differential $k$-form $\omega$ to the $(k+1)$-form $d \omega$  that on an infinitesimal $(k+1)$-simplex $(x_0, x_1, \cdots, x_{k+1})$ in $X$ evaluates to

$$
  d\omega(x_0, x_1, \cdots, x_{k+1})
  :=
  \sum_{i=0}^{k+1}
  \omega(x_0, \cdots , \hat{x_i}, \cdots, x_{k+1})
  \,,
$$

where the hat indicates that the corresponding variable is omitted, as usual.

We recognize this as the alternating sum of the [[simplicial identities|face maps]]
$\partial_i^*$
of the [[simplicial object|cosimplicial object]] $C^\infty(X^{\Delta_{inf}^\bullet})$.

$$
  d := \sum_{i=0}^{k+1} \partial_i^* :
  \Omega^k(X) \to \Omega^{k+1}(X)
  \,.
$$

These constructions remind one and should be compared with the [[Dold-Kan correspondence]]. In particular with its _dual_ (cosimplicial) version as recalled in section 4 of [CastiglioniCortinas](http://arxiv.org/PS_cache/math/pdf/0306/0306289v3.pdf)

In total this should show the following

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

Let $X$ be a synthetic differential space and $C^\infty(X^{\Delta_{inf}^\bullet})$ the [[simplicial object|cosimplicial object]] of [[generalized smooth algebra]]s of functions on the spaces of infinitesimal $k$-simplices in $X$.

Then the deRham complex $(\Omega^\bullet(X), d)$ of differential forms on $X$ is the normalized [[Moore complex]] of the cosimplicial object $C^\infty(X^{\Delta_{inf}^\bullet})$.
=--

In other words, in as far as the Dold-Kan correspondence is an equivalence, we find that:

the object of differential forms on $X$ _is_ the cosimplicial [[generalized smooth algebra]] $C^\infty(X^{\Delta_{inf}^k})$.

## References

* {#Kock10} [[Anders Kock]], _Synthetic geometry of manifolds_, Cambridge Tracts in Mathematics **180** (2010) [[pdf](http://home.imf.au.dk/kock/SGM-final.pdf), [doi:10.1017/CBO9780511691690](https://doi.org/10.1017/CBO9780511691690)]

* [[Ieke Moerdijk]], [[Gonzalo Reyes|Gonzalo E. Reyes]]: *[[Models for Smooth Infinitesimal Analysis]]*, Springer (1991) [[doi:10.1007/978-1-4757-4143-8](https://link.springer.com/book/10.1007/978-1-4757-4143-8)]

* J. L Castiglioni, G. Cortiñas, _Cosimplicial versus DG-rings: a version of the Dold-Kan correspondence_, J. Pure Applied Algebra **191** (2004) 119-142 [[arXiv:math/0306289](https://arxiv.org/abs/math/0306289), [doi:10.1016/j.jpaa.2003.11.009](https://doi.org/10.1016/j.jpaa.2003.11.009)]


differential forms on presheaves

#Contents#
* table of contents
{:toc}

## Idea

Let $C$ be any [[small category]], write $PSh(C) = [C^{op}, Set]$ for its [[category of presheaves]] and let

$$
  \Omega^\bullet_C : C^{op} \to dgAlg
$$

be _any_ functor to the category of [[dg-algebra]]s. Following the logic of [[space and quantity]], we may think of the objects of $C$ as being _test spaces_ and the functor $\Omega^\bullet_C$ as assigning to each test space its [[deRham complex|deRham dg-algebra]].

An example of this construction that is natural from the point of view of [[differential geometry]] appears in the study of [[diffeological space]]s, where $C$ is some subcategory of the category [[Diff]] of smooth [[manifold]]s, and $\Omega^\bullet_C$ is the restriction of the ordinary assignment of [[differential form]]s to this. But in the application to topological spaces, in the following, we need a choice for $C$ and $\Omega^\bullet_C$ that is non-standard from the point of view of [[differential geometry]]. Still, it follows the same general pattern.

After postcomposing with the [[forgetful functor]] that sends each [[dg-algebra]] to its underlying [[set]], the functor $\Omega^\bullet_C$ becomes itself a [[presheaf]] on $C$. For $X \in PSh(C)$ any other presheaf, we extend the notation and write

$$
  \Omega^\bullet_C(X) := Hom_{PSh(C)}(X, \Omega^\bullet_C)
$$

for the [[hom-set]] of presheaves. One checks that this set naturally inherits the structure of a [[dg-algebra]] itself, where all operations are given by applying "pointwise" for each $p : U \to X$ with $U \in C$ the operations in $\Omega^\bullet_C(U)$. This way we get a functor

$$
  \Omega^\bullet_C : PSh(C) \to dgAlg^{op}
$$

to the [[opposite category]] of that of [[dg-algebra]]s. We may think of $\Omega^\bullet_C(X)$ as the deRham complex of the presheaf $X$ as seen by the functor $\Omega^\bullet_C : C \to dgAlg^{op}$.

By [[category theory|general abstract nonsense]] this functor has a [[right adjoint]] $K_C : dgAlg^{op} \to PSh(C)$, that sends a dg-algebra $A$ to the [[presheaf]]

$$
  K_C(A) : U \mapsto Hom_{dgAlg}(\Omega^\bullet_C(U), A)
  \,.
$$

The [[adjunction]]

$$
  \Omega^\bullet_C : PSh(C) \stackrel{\leftarrow}{\to} : dgAlg^{op} : K_C
$$

is an example for the adjunction induced from a [[dualizing object]].

## Examples

### Differential forms on simplicial sets

There are various variant of [[differential forms on simplices]]. Each gives rise to a notion of differential forms on [[simplicial set]]s. This is also known as the [[Sullivan construction]] in [[rational homotopy theory]].

[[!redirects differential forms on simplicial sets]]


differential forms on simplices

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Rational homotopy theory
+--{: .hide}
[[!include differential graded objects - contents]]
=--
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

There are various [[simplicial object|simplicial]] [[dg-algebra]]s that assign to the standard $n$-[[simplex]] a kind of [[de Rham algebra]] on $\Delta^n$.

By the discussion at [[differential forms on presheaves]], each such extends to a notion of differential forms on simplicial sets.

## Definition

### Smooth differential forms

+-- {: .num_defn #SmoothnSimplex}
###### Definition
**(smooth $n$-simplex)**

For $n \in \mathbb{N}$ the _smooth [[n-simplex]]_ $\Delta^n_{smth}$ is the [[smooth manifold]] with [[manifold with boundary|boundary]] and [[manifold with corners|corners]] defined, up to [[isomorphism]], as the following locus inside the [[Cartesian space]] $\mathbb{R}^{n+1}$:

$$
  \Delta^n_{smth}
  \;\coloneqq\;
  \Big\{
    (x_0, x_1, \cdots, x_n)
    \in
    \mathbb{R}^{n+1}
     \;\vert\;
     0 \leq x_i \leq 1
     \;\text{and}\;
     \underoverset{i = 0}{n}{\sum}
     x_i
     \; = 1
  \Big\}
  \hookrightarrow
  \mathbb{R}^{n+1}
  \,.
$$

For $0 \leq i \leq n$ the [[function]]

$$
  x_i \;\colon\; \Delta^n_{smth} \to \mathbb{R}
$$

which picks the $i$th component in the above definition is called the $i$th _barycentric coordinate function_.

For

$$
  f \;\colon\; [n_1] \longrightarrow [n_2]
$$

a morphism of finite non-empty [[linear orders]] $[n] \coloneqq \{0 \lt 1 \lt \cdots \lt n\}$, let

$$
  \Delta_{smth}(f)
     \;\colon\;
  \Delta^{n_1}_{smth}
    \longrightarrow
  \Delta^{n_2}_{smth}
$$

be the [[smooth function]] defined by $x_i \mapsto x_{f(i)}$.

=--

+-- {: .num_defn #SmoothDifferentialFormsOnSmoothnSimplex}
###### Definition
**(smooth differential forms on the smooth $n$-simplex)

For $k \in \mathbb{N}$ then a [[smooth differential k-form]] on the smooth $n$-simplex (def. \ref{SmoothnSimplex}) is a smooth differential form in the sense of [[smooth manifolds]] with [[manifold with boundary|boundary]] and [[manifold with corners|corners]]. Explicitly this means the following.

Let

$$
  F^n
  \;\coloneqq\;
  \left\{
    (x_0, x_1, \cdots, x_n)
    \in
    \mathbb{R}^{n+1}
     \;\vert\;
     \underoverset{i = 0}{n}{\sum}
     x_i
     \; = \; 1
  \right\}
  \hookrightarrow
  \mathbb{R}^{n+1}
$$

be the affine plane in $\mathbb{R}^{n+1}$ that contains $\Delta^n_{smth}$ in its defining inclusion from def. \ref{SmoothnSimplex}. This is a [[smooth manifold]] [[diffeomorphism|diffeomorphic]] to the [[Cartesian space]] $\mathbb{R}^{n}$.

A smooth differential form on $\Delta^n_{smth}$ of degree $k$ is a collection of [[linear functions]]

$$
  \wedge^k T_x F^n \longrightarrow \mathbb{R}
$$

out of the $k$-fold skew-symmetric [[tensor power]] of the [[tangent space]] of $F^n$ at some point $x$ to the [[real numbers]], for all $x \in \Delta^n_{smth}$ such that this extends to a smooth differential $k$-form on $F^n$.

Write $\Omega^\bullet(\Delta^n_{smth})$ for the graded [[real vector space]] defined this way. By definition there is then a canonical linear map

$$
  \Omega^\bullet(F^n) \longrightarrow \Omega^\bullet(\Delta^n_{smth})
$$

from the [[de Rham complex]] of $F^n$ and there is a unique structure of a [[differential graded-commutative algebra]] on $\Omega^\bullet(\Delta^n_{smth})$ that makes is a [[homomorphism]] of [[dg-algebras]] form the [[de Rham algebra]] of $F^n$. This is the de Rham algebra of smooth differential forms on the smooth $n$-simplex.

For $f \colon [n_1] \to [n_2]$ a homomorphism of finite non-empty [[linear orders]] with $\Delta_{smth}(f) \colon \Delta^{n_1}_{smth} \to \Delta^{n_2}_{smth}$ the corresponding smooth function according to def. \ref{SmoothnSimplex}, there is the induced [[homomorphism]] of [[differential graded-commutative algebras]]

$$
  (\Delta_{smth}(f))^\ast
    \;\colon\;
  \Omega^\bullet(\Delta^{n_2}_{smth})
    \longrightarrow
  \Omega^\bullet(\Delta^{n_1}_{smth})
$$

induced from the usual [[pullback of differential forms]] on $F^n$. This makes smooth differential forms on smooth simplices be a [[simplicial object]] in [[differential graded-commutative algebras]]:

$$
  \Omega^\bullet(\Delta^{(-)}_{smth})
    \;\colon\;
   \Delta^{op}
     \longrightarrow
   dgcAlg_{\mathbb{R}}
  \,.
$$

=--

The standard proof of the [[Poincaré lemma]] applies to show that

$$
  H^\bullet(\Omega^\bullet(\Delta^n_{smth})) \simeq \mathbb{R}
  \,.
$$

Each element of $\Omega^p_{poly}(\Delta^n)$ may be uniquely written

$$\Phi =\sum_{1\leq i_1\lt\ldots\lt i_p\leq n}\Phi_{i_1\ldots i_p}d b_{i_1}\wedge\ldots d b_{i_p},$$

where $b_j$ is as above the $j^{th}$ barycentric coordinate function and each $\Phi_{i_1\ldots i_p}$ is a $C^\infty$-function on $\mathbf{\Delta}^n$.

With this representation the multiplication and differential are given by the usual formulae. The multiplication is defined by $\Phi \wedge \Psi$ and extends linearly the product

$$(d b_{i_1}\wedge\ldots d b_{i_p})\wedge (d b_{j_1}\wedge\ldots d b_{j_q}) = (d b_{i_1}\wedge\ldots d b_{i_p}\wedge d b_{j_1}\wedge\ldots d b_{j_q})$$

on the generating forms. Now if $f$ is a differentiable function

$$d f = \sum_{i=1}^{n} \frac{\partial f}{\partial x^i}d x_i,$$

so if

$$\Phi =\sum_{1\leq i_1\lt \ldots\lt i_p\leq n}\Phi_{i_1\ldots i_p}d b_{i_1}\wedge\ldots d b_{i_p},$$

then

$$d\Phi =\sum_{1\leq i_1\lt\ldots\lt i_p\leq n} d\Phi_{i_1\ldots i_p} \wedge d b_{i_1} \wedge \ldots d b_{i_p},$$

### Polynomial differential forms
  {#Polynomial}

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

For $n \in \mathbb{N}$ write

$$
  \Omega_{poly}^{\bullet}(\Delta^n)
   \;\coloneqq\;
  Sym^\bullet_{\mathbb{Q}}
   \langle t_0, \cdots, t_n, d t_0, \cdots, d t_n\rangle/\left(\sum t_i -1, \sum d t_i \right)
$$

for the [[quotient]] of the $\mathbb{Z}$-graded [[symmetric algebra]] over the [[rational numbers]] on $n+1$ generators $t_i$ in degree 0 and $n+1$ generators $d t_i$ of degree 1.

In particular, in degree 0 these are called the _polynomial functions_

$$
  \Omega^0_{poly}(\Delta^n)
  \;=\;
  \mathbb{Q}[t_0, t_1, \cdots t_n]/\left( \underset{i}{\sum} t_i = 0 \right)
$$

due to the canonical inclusion

$$
  \Omega^0_{poly}(\Delta^n)
    \hookrightarrow
  C^\infty(\Delta^n_{smth})
$$

into the [[smooth functions]] on the $n$-simplex according to def. \ref{SmoothDifferentialFormsOnSmoothnSimplex}, obtained by regarding the generator $t_i$ as the $i$th barycentric coordinate function.

Observe that the [[tensor product]] of the polynomial differential forms over these polynomial functions with the [[smooth functions]] on the $n$-simplex, is canonically [[isomorphism|isomorphic]] to the space $\Omega^\bullet(\Delta^n_{smth})$ of smooth [[differential forms]], according to def. \ref{SmoothDifferentialFormsOnSmoothnSimplex}:

$$
  \Omega^\bullet(\Delta^n_{smth})
    \simeq
  C^\infty(\Delta^n_{smth}) \otimes_{\Omega^0_{poly}(\Delta^n)}
  \Omega^\bullet_{poly}(\Delta^n)
$$

where moreover the generators $d t_i$ are identified with the de Rham differential of the $i$th barycentric coordinate functions.

This defines a canonical inclusion

$$
  \Omega^\bullet_{poly}(\Delta^n)
    \hookrightarrow
  \Omega^\bullet(\Delta^n_{smth})
$$

and there is uniquely the structure of a [[differential graded-commutative algebra]] on $\Omega^\bullet_{poly}(\Delta^n)$ that makes this a [[homomorphism]] of [[dg-algebras]]. This is the _dg-algebra of polynomial differential forms_.

For $f \colon [n_1] \to [n_1]$ a [[morphism]] of finite non-empty [[linear orders]], let

$$
  \Omega^\bullet_{poly}(f)
   \;\colon\;
  \Omega^\bullet_{poly}(\Delta^{n_2}) \to \Omega^\bullet_{poly}(\Delta^{n_1})
$$

be the morphism of dg-algebras given on generators by

$$
  \Omega^\bullet_{poly}(f) : t_i \mapsto \sum_{f(j) = i} t_j
  \,.
$$

This yields a [[simplicial object|simplicial]] [[differential graded-commutative algebra]]

$$
  \Omega^\bullet_{poly}(\Delta^{(-)}) : \Delta^{op} \to cdgAlg_k
$$

which is a sub-simplicial object of that of smooth differential form

$$
  \Omega^\bullet_{poly}(\Delta^{(-)})
    \hookrightarrow
  \Omega^\bullet(\Delta_{smth}^{(-)})
  \,.
$$

=--

### Piecewise polynomial differential forms
 {#PiecewisePolynomialDifferentialForms}

By left [[Kan extension]] the functor of polynomial differential forms from def. \ref{PolynomialDifferentialForms} yields a functor on all [[simplicial sets]]

$$
  \Omega^\bullet_{poly} \colon sSet \longrightarrow cdgAlg_k^{op}
  \,.
$$

This is the [[left adjoint]] in a [[nerve and realization]] [[adjunction]]

$$
  (\Omega^\bullet_{poly} \dashv \mathcal{K}_{poly})
    \;\colon\;
  (dgcAlg_{\mathbb{Q}, \geq 0})^{op}
    \underoverset
      {\underset{K_{poly}}{\longrightarrow}}
      {\overset{\Omega^\bullet_{poly}}{\longleftarrow}}
      {\bot}
  sSet
  \,.
$$

Composing with the [[singular simplicial complex]] functor

$$
  Sing \;\colon\; Top \longrightarrow sSet
$$

on [[topological spaces]], this yields a functor on topological spaces

$$
  \Omega^\bullet_{pwpoly}
   \;\colon\;
  Top
   \overset{Sing}{\longrightarrow}
  sSet
   \overset{\Omega^\bullet_{poly}}{\longrightarrow}
  (dgcAlg_{\mathbb{Q}, \geq 0})^{op}
$$

which we may think of as assigning "piecewise polynomial" differential forms.

This is the starting point of the Sullivan approach to [[rational homotopy theory]]. See there for more

## Properties {#Properties}

Let $k$ be a [[field]] of [[characteristic]] 0. Let $\Omega^\bullet_{poly} : sSet \to cdgAlg_k^{op}$ be the left [[Kan extension]] of $\Omega^\bullet_{poly} : \Delta \to cdgAlg_k^{op}$ from [above](#Polynomial).

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

For $S \in sSet$,  define a morphism of graded $k$-vector spaces

$$
  \int : \Omega^\bullet_{poly}(S) \to C^\bullet(S, k)
$$

from polynomial differential forms on simplices to [[cochains on simplicial sets]] by sending $\omega \in \Omega^n_{poly}(K)$ to the cochain that sends $\sigma \in K_n$ to

$$
  \int_\sigma f := \int_{\Delta^n} f_{max}(\sigma) d t_1 \wedge \cdots d t_n
  \,,
$$

where on the right we have the ordinary [[integral]] of the $1,\cdots,n$-component of the restriction of $f$ to $\sigma$.

=--

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

The morphism $\int$ is a [[quasi-isomorphism]] of cochain complexes.

=--

This is ([Bousfield-Gugenheim, theorem 2.2, corollary 3.4](#BousfieldGugenheim)).

The following is the central fact of the Sullivan approach to [[rational homotopy theory]]:

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

The functor $\Omega^\bullet_{poly}$ is the [[left adjoint]] of a [[Quillen adjunction]]

$$
  (\Omega^\bullet_{poly} \dashv R)
  \;\colon\;
  sSet
    \underoverset
      {\underset{\Omega^\bullet_{poly}}{\to}}
      {\overset{R}{\leftarrow}}
      {\bot}
  cdgAlg_k^{op}
$$

for the standard [[model structure on simplicial sets]] and the projective [[model structure on dg-algebras|model structure on commutative dg-algebras]].

=--

This is shown in ([Bousfield-Gugenheim, section 8](#BousfieldGugenheim)).

So in particular $\Omega^\bullet_{poly}$ sends cofibrations of simplicial sets to fibrations of dg-algebras. Hence for $i : \partial \Delta[k] \hookrightarrow \Delta[k]$ a boundary inclusion the corresponding restriction

$$
  i^* : \Omega^\bullet_{poly}(\Delta^k) \to \Omega^\bullet_{poly}(\partial \Delta^k)
$$

is degreewise surjective.

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

The functor $\Omega^\bullet_{poly}$ is a [[lax monoidal functor]] whose lax monoidal structure map

$$
  \nabla_{X,Y}
  :
  \Omega^\bullet_{poly}(X)
  \otimes
  \Omega^\bullet_{poly}(Y)
  \to
  \Omega^\bullet_{poly}(X \times Y)
$$

is a [[quasi-isomorphism]].

=--

This is reviewed for instance in ([Hess, page 12](#Hess)).

## Applications

Applications include

* the [[Sullivan construction]] in [[rational homotopy theory]];

* the [[sSet]]-[[enriched category|enrichment]] of the [[model structure on dg-algebras over an operad]].

* higher [[Lie integration]]

## Related concepts

[[PL de Rham complex]]

## References

Original articles:

* [[Dennis Sullivan]], p. 297 (30 of 64) of: _Infinitesimal computations in topology_, Publications Mathématiques de l'IHÉS, 47 (1977), p. 269-331 ([numdam:PMIHES_1977__47__269_0](http://www.numdam.org/item/PMIHES_1977__47__269_0))

* {#BousfieldGugenheim} [[Aldridge Bousfield]], [[Victor Gugenheim]], p. 1 of: _[[On PL deRham theory and rational homotopy type]]_, Memoirs of the AMS, vol. 179 (1976) ([ams:memo-8-179](https://bookstore.ams.org/memo-8-179))

* {#GriffithMorgan13} [[Phillip Griffiths]], [[John Morgan]], p. 83-84 of: _Rational Homotopy Theory and Differential Forms_,  Progress in Mathematics Volume 16, Birkhauser (2013) ([doi:10.1007/978-1-4614-8468-4](https://doi.org/10.1007/978-1-4614-8468-4))

Review:

* {#FelixHalperinThomas00} [[Yves Félix]], [[Stephen Halperin]], [[Jean-Claude Thomas]], p. 122 (154 of 574) in: _Rational Homotopy Theory_, Graduate Texts in Mathematics, 205, Springer-Verlag, 2000 ([doi:10.1007/978-1-4613-0105-9](https://link.springer.com/book/10.1007/978-1-4613-0105-9))

* {#Hess06} [[Kathryn Hess]], Def. 1.19 (p. 8) of:  _Rational homotopy theory: a brief introduction_, contribution to _[Summer School on Interactions between Homotopy Theory and Algebra](https://jdc.math.uwo.ca/summerschool/)_, University of Chicago, July 26-August 6, 2004, Chicago ([arXiv:math.AT/0604626](http://arxiv.org/abs/math.AT/0604626)), chapter in Luchezar Lavramov, [[Dan Christensen]], [[William Dwyer]], [[Michael Mandell]], [[Brooke Shipley]] (eds.), _Interactions between Homotopy Theory and Algebra_, Contemporary Mathematics 436, AMS 2007 ([doi:10.1090/conm/436](http://dx.doi.org/10.1090/conm/436))

[[!redirects polynomial differential form]]
[[!redirects polynomial differential forms]]

[[!redirects polynomial differential forms on simplices]]

[[!redirects piecewise polynomial differential form]]
[[!redirects piecewise polynomial differential forms]]

[[!redirects polynomial differential forms on the n-simplex]]
[[!redirects polynomial differential forms on n-simplices]]

[[!redirects smooth differential forms on simplices]]

[[!redirects Sullivan differential form]]
[[!redirects Sullivan differential forms]]


differential forms on supermanifolds > history

< [[differential forms on supermanifolds]]
[[!redirects differential forms on supermanifolds -- history]]


differential function complex

> under construction

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Differential cohomology
+--{: .hide}
[[!include differential cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _differential function complex_ ([HopkinsSinger](#HopkinsSinger)) is a [[Kan complex]] of [[cocycle]] s for _generalized_ [[differential cohomology]], hence for differential refinements of [[generalized (Eilenberg-Steenrod) cohomology]] theories:

roughly, given a [[spectrum]] $E$ [[Brown representability|representing]] a given [[cohomology theory]], its _differential function complex_ over any given [[smooth manifold]] $U$ is the [[simplicial set]] whose $k$-[[simplices]] are triples consisting of

* a [[continuous function]] $f : U \times \Delta^k \to E_{n}$;

* a smooth [[differential form]] $\omega$ on $U \times \Delta^k$ whose corresponding real cohomology class (under the [[de Rham theorem]]) is that of the pullback of the real cohomology classes of $E$ along $f$;

* an explicit [[coboundary]] in real cohomology exhibiting this fact.

(More precisely, in order for this construction to yield not just a single [[simplicial set]] (which will be a [[Kan complex]]) but a suitable [[spectrum]] object, there are conditons on the dependency of $\omega$ on the [[tangent vector]]s to the [[simplex]].)

When applied to the [[Eilenberg-MacLane spectrum]] $K\mathbb{Z}$ this construction reproduces, on cohomology classes, [[ordinary differential cohomology]]. Applied to the [[classifying space]] $B U$ of [[topological K-theory]] it gives [[differential K-theory]].

See also at _[differential cohomology diagram --Hopkins-Singer coefficients](differential+cohomology+diagram#HopkinsSingerCoefficients)_.

## Definitions

### Cocycles with values in graded vector spaces

For the present purposes it will be convenient to collect [[cocycle]]s of various degrees together to a single cocycle. For that purpose we make the following simple definition.

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

For $V = V^\bullet$ a [[graded vector space]] over the [[real number]]s set

* for $E$ a [[topological space]]:

  $$
    C^\bullet(E, V)^n := \oplus_{i + j = n} C^i(E, V^j)
  $$

* and so on

=--

(...)

### Differential functions

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

For

* $E$ a [[topological space]];

* $\iota \in Z^n(E,\mathbb{R})$ a [[cocycle]] on $E$ for real-valued [[singular cohomology]] on $E$,

a **differential function** on a smooth [[manifold]] $U$ with values in $(E,\iota)$ is a triple $(c,h,\omega)$ with

* $c : U \to E$ a continuous map;

* $\omega \in \Omega^n(S)$ a smooth [[differential form]] on $S$;

* $h \in C^{n-1}(U,\mathbb{R})$ a cochain in real cohomology on (the topological space underlying) $U$;

such that in the [[abelian group]] $Z^n(S,\mathbb{R})$ of singular cochains the equation

$$
  \omega = c^*\iota + \delta h
$$

holds, where

* $\omega$ is here regarded  as a singular cochain (that sends a chain to the integral of $\omega$ over it, as discussed at [[de Rham theorem]]),

* $\delta$ denotes the coboundary operator,(the [[Moore complex]] differential of the [[singular simplicial complex]]).

=--

This is ([HopkinsSinger, def.4.1](#HopkinsSinger)).

In words this is: a continuous map to the topological space together with a _smooth_ refinement of the pullback of the chosen singular cochain.

### Differential function complexes

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

For

* $E$ be a [[topological space]] and

* $\iota \in Z^n(E,\mathbb{R})$ a [[cocycle]] on $E$ for real-valued [[singular cohomology]] on $X$,

* $U$ a [[smooth manifold]],

the **differential function complex**

$$
  (E,\iota)^U
$$

of all differential functions $S \to (X,\iota)$ is the [[simplicial set]] whose $k$-[[simplices]] are differential functions, def, \ref{DifferentialFunctions}

$$
  U \times \Delta^k_{Top} \to (E,\iota)
  \,.
$$

=--

For applications one needs certain sub-complex of this, filtered by the number of legs that $\omega$ has along the
simplices.

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

For $s \in \mathbb{N}$ write

* $filt_s \Omega^\bullet(U \times \Delta^k)$

  for the sub-simplicial set of differential forms that vanish when evaluated on more than $s$ [[vector field]]s tangent to the [[simplex]];

* $filt_s (X,\iota)^S \subset (X,\iota)^S$

  for the sub-simplicial set of those differential functions whose differential form component is in $filt_s \Omega^\bullet(U \times \Delta^k)$.

=--

This is ([HopkinsSinger, def. 4.5](#HopkinsSinger)).

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

The complex $filt_s (E,\iota)^U$ is (up to [[weak homotopy equivalence|equivalence]], of course) the [[homotopy pullback]]

$$
  \array{
    filt_s (E,\iota)^U &\to&
      filt_s \Omega^n_{cl}(U \times \Delta^\bullet, \mathcal{V})
    \\
    \downarrow && \downarrow
    \\
    Sing E^U &\to& Z^\bullet(U \times \Delta^\bullet, \mathcal{V})
  }
$$

in [[sSet]] (regarded as equipped with its standard [[model structure on simplicial sets]]).

=--

Here $E^U$ is the [[internal hom]] in [[Top]] and $Sing(-)$ denotes the [[singular simplicial complex]].

The following proposition gives the [[simplicial homotopy groups]] of these differential function complexes in dependence of the parameter $s$.

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

We have generally

$$
  \pi_k Z(S \times \Delta^\bullet_{Diff}, \mathcal{V}) =
   H^{n-m}(S; \mathcal{V})
$$

(for instance by the [[Dold-Kan correspondence]]).

The [[simplicial homotopy group]]s of $filt_s \Omega^n_{cl}(S \times \Delta^\bullet_{Diff})$ are

$$
  \pi_k filt_s \Omega^n_{cl}(S \times \Delta^\bullet_{Diff})
  =
  \left\{
    \array{
       H_{dR}^{n-k}(S, \mathcal{V}) & | k \lt s
       \\
       \Omega_{cl}(S; \mathcal{V})^{n-s} & | k = s
       \\
       0 & | k \gt s
    }
  \right\}
  \,.
$$

This implies [[isomorphism]]s

$$
  \pi_k filt_s(X; \iota)^S
   \stackrel{\simeq}{\to}
  \left\{
    \array{
       \pi_k X^S & | k \lt s
       \\
       H^{n-k-1}(S; \mathcal{V})/ \pi_{k+1} X^S
       |
       k \gt s
    }
  \right.
  \,.
$$

=--

This appears as [HopkinsSinger, p. 36 and corollary D15](#HopkinsSinger).

### Differential $E$-cohomology

Let $E_\bullet$ be an [[Omega-spectrum]]. Let $\iota_\bullet$ be the canonical [[Chern character]] class (...).

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

For $S$ a [[smooth manifold]], and $s \in \mathbb{N}$, the sequence of differential function complexes, def. \ref{DifferentialFunctionComplex},

$$
  filt_{s + n}(E_n; \iota_n)^S
   \stackrel{\simeq}{\to}
  \Omega  filt_{s + (n + 1)}(E_{n+1}; \iota_{n+1})^S
$$

forms an [[Omega-spectrum]].

This is the **differential function spectrum** for $E$, $S$, $s$.

=--

This is ([HopkinsSinger, section 4.6]).

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

The **differential $E$-cohomology group** of the smooth manifold $S$ in degree $n$ is

$$
  H_{diff}^n(S,E) :=
  \pi_0
  filt_0(E_n \iota_n)^S
$$

=--

This is ([HopkinsSinger, def. 4.34](#HopkinsSinger)).

## Properties

### Homotopy groups

For reference, we repeat from above the central statements about the [[homotopy type]]s of the differential function complexes, def. \ref{DifferentialFunctionComplex}.

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

For $E$ an [[Omega-spectrum]], $S$ a [[smooth manifold]],
we have for all $s,n \in \mathbb{N}$, a [[weak homotopy equivalence]]

$$
  \Omega  filt_{s+1}(E_{n}; \iota_{n})^S
   \stackrel{\simeq}{\to}
  filt_{s}(E_{n-1}; \iota_{n-1})^S
  \,,
$$

identifying the [[loop space object]] (at the canonical base point) of the differential function complex of $E_{n}$ at filtration level $s+1$ with that differential function complex of $E_{n-1}$ at filtration level $s$.

=--

### Relation to differential cohomology in cohesive $(\infty,1)$-toposes

The following is a simple corollary or slight rephrasing of some of the above constructions, which may serve to show how differential function complexes present differential cohomology in the [[cohesive (∞,1)-topos]] of [[smooth ∞-groupoid]]s.

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

For $E_\bullet$ a spectrum as above,
we have an [[(∞,1)-pullback]] square

$$
  \array{
    filt_0 (E_n; \iota_n)^{(-)} &\to& \prod_i \Omega^{n_i}_{cl}(-)
    \\
    \downarrow && \downarrow
    \\
    Disc E_n
    &
     \stackrel{}{\to}
    &
    \prod_i \mathbf{B}^{n_i} \mathbb{R}_{disc}
  }
  \,.
$$

=--

+-- {: .proof}
###### Proof

By prop. \ref{HomotopyGroupsOfDiffFunctionComplex} we have that

* $filt_\infty (E; \iota_n)^S \simeq Sing X^S$;

* $filt_0 \Omega_{cl}(S \times \Delta^\bullet) \simeq \Omega_{cl}(S)$.

The statement then follows with the [[pasting law]] for [[homotopy pullback]]s

$$
  \array{
     filt_0 (E_n; \iota_n)^S &\to&  \Omega^n_{cl}(S; \mathcal{V})
     \\
     \downarrow && \downarrow
     \\
     filt_\infty (E_n; \iota_n)^S &\to&  filt_\infty \Omega_{cl}(S \times \Delta^\bullet; \mathcal{V})
     \\
     \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}}
     \\
     Sing X^S &\to& Z(S \times \Delta^\bullet; \mathcal{V})
  }
  \,.
$$

(...)

=--

## Examples

* For $E = H \mathbb{Z}$ the [[Eilenberg-MacLane spectrum]], $H_{diff}^n(-,E)$ is [[ordinary differential cohomology]].

* For $E = K U$ the [[K-theory spectrum]], $H_{diff}^n(-,E)$ is [[differential K-theory]].

* For $E = M O, M U$ the [[Thom spectrum]], $H_{diff}^n(-,E)$ is [[differential cobordism cohomology]];

* For $E = tmf$ the [[tmf]] spectrum, $H_{diff}^n(-,E)$ is [[differential cohomology]];

### Line bundles with connection

Let $X = \mathcal{B} U(1) \simeq K(\mathbb{Z},2)$ be the [[Eilenberg-MacLane space]] that is the [[classifying space]] for $U(1)$-[[principal bundle]]s. It carries the canonical [[cocycle]] $\iota := Id : \mathcal{B}U(1) \to \mathcal{B}U(1) \simeq K(\mathbb{Z},2)$ representing in $H^2(X,\mathbb{Z})$ the class of the universal complex [[line bundle]] $L \to X$ on $X$.

Accordingly, for $c : S\to \mathcal{B}U(1)$ a continuous map, we have the corresponding line bundle $c^* L$ on $S$.

One checks (...details...Example 2.7 in HopSin) that a refinement of $c$ to a differential function $(c,\omega,h)$ corresponds to equipping $c^* L$ with a [[connection on a bundle|smooth connection]].

Now consider $((c,\omega,h) \to (c',\omega', h')) \in filt_0  (\mathcal{B}U(1),Id)^S$ a morphism between two such $(\mathcal{B}U(1),Id)$-differential functions. By definition this is now a $U(1)$-principal bundle $\hat L$ with connection on $S \times \Delta^i_{Diff}$, whose curvature form $\hat \omega \in \Omega^2(S \times \Delta^1_{Diff})$ is of the form $g \cdot \tilde \omega$, where $\tilde \omega$ is a 2-form on $S$ and $g$ is a smooth function on $\Delta^1_{Diff}$, both pulled back to $S \times \Delta^1_{Diff}$ and multiplied there.

But since $\hat \omega$ is necessarily _closed_ it follows with $d (g \wedge \tilde \omega) = d t \frac{\partial g}{\partial t} \wedge \tilde \omega + g \wedge d_{S} \tilde \omega$ that $g$ is actually constant.

This means that that the parallel transoport of the connection $\hat \nabla$ on $S \times \Delta^1_{Diff}$ induces a insomorphism between the two line bundles on $S$ over the endpoints of $S \times \Delta^1_{Diff}$ that respects the connections.

...

### Differential K-cocycles

(...)

### Higher filtration degree

+-- {: .num_example #HigherFilteringOfOrdinaryDiffCoh}
###### Example

For $E = H \mathbb{Z}$ the [[Eilenberg-MacLane spectrum]], prop. \ref{LoopObjectsOfDiffFunctionComplexes} states that
$filt_{1}(H \mathbb{Z}^{n+1}; \iota_{n})^S$ is an [[n-groupoid]] such that the [[automorphism]]s of the 0-object form [[ordinary differential cohomology]] in degree $n$.

$$
  \Omega filt_{1}(H \mathbb{Z}^{n+1}; \iota_{n})^S
  \simeq
  \mathbf{H}_{diff}^n(S)
  \,.
$$

=--

+-- {: .num_remark}
###### Remark

Example \ref{HigherFilteringOfOrdinaryDiffCoh} for $n = 4$
plays a central role in the description of [[T-duality]] by
[[twisted differential K-theory]] in ([KahleValentino](#KahleValentino)).

=--

## References

Differential function complexes were introduced and studied in

* [[Mike Hopkins]], I. Singer, _[[Quadratic Functions in Geometry, Topology,and M-Theory]]_
 {#HopkinsSinger}

For further references see [[differential cohomology]].

* [[Alexander Kahle]], [[Alessandro Valentino]], _[[T-Duality and Differential K-Theory]]_
  {#KahleValentino}

[[!redirects differential function complexes]]


differential Galois theory

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## Idea

_Differential Galois theory_ is an analogue of [[Galois theory]] where [[fields]] are generalized to [[differential fields]], hence is a theory for [[differential equations]] rather than just algebraic [[equations]].

## References

* [[Pierre Deligne]], _[[Catégories Tannakiennes]]_, Grothendieck Festschrift, vol. II, Birkhäuser Progress in Math. 87 (1990) pp.111-195.

* Andy Magid, _Differential Galois theory_, Notices of the AMS (1999) [pdf](http://www.ams.org/notices/199909/fea-magid.pdf)

* Teresa Crespo, _The origins of differential Galois theory_ ([pdf slides](http://atlas.mat.ub.es/personals/crespo/Origins.pdf))

* Andy R. Magid, _Universal covers and category theory in polynomial and differential Galois theory_, [pdf](http://www.math.ou.edu/~amagid/fieldsproc.pdf)

Textbook accounts include

* Marius van der Put, Michael Singer, _Galois theory of linear differential equations_, Springer, Berlin (2003)

* M. Singer, _Direct and inverse problems in differential Galois theory_, in H. Bass, et al. (eds.) _Selected works of Ellis Kolchin with commentary_ , Amer. Math. Soc. 1999,  527–554.

Discussion in terms of [[D-geometry]] is in

* [[João Pedro P. dos Santos]], _The behaviour of the differential Galois group on the generic and special fibres: A Tannakian approach_,  J. reine angew. Math. 637 (2009), 63-98 ([pdf](http://www.math.jussieu.fr/~dos-santos/BDGG_gen_special_V2.pdf))

* [[João Pedro P. dos Santos]], _Lifting D-modules from positive to zero characteristic_ ([pdf](http://www.math.jussieu.fr/~dos-santos/V2_22_2_2010.pdf))

based on

* B. H. Matzat, M. van der Put, _Iterative differential equations and the Abhyankar conjecture_, J. Reine Angew. Math. 557 (2003), pp. 1–52.

A connection to [[geometric invariant theory]] is proposed in

* [[Yves Andre]], _Solution algebras of differential equations and quasi-homogeneous varieties_, [arxiv/1107.1179](http://arxiv.org/abs/1107.1179)

[[model theory|Model theoretic]] aspects are discussed in

* Moshe Kamensky, [[Anand Pillay]], _Interpretations and differential Galois extensions_, [arxiv/1409.5915](http://arxiv.org/abs/1409.5915)
* [[Anand Pillay]], _Differential Galois theory I_, Illinois J. Math. __42__(4) 1998
* D. Bertrand, A. Pillay, _Galois theory, functional Lindemann-Weierstrass, and Manin maps_, Pacific J. Math. 281:1 (2016) 51–82 [doi](https://doi.org/10.2140/pjm.2016.281.51)

[[!redirects differential Galois group]]


differential geometry

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#### Higher geometry
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#Contents#
* table of contents
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## Scope

__Differential geometry__ is a mathematical discipline studying geometry of spaces using differential and integral calculus. Classical differential geometry studied submanifolds (curves, surfaces...) in Euclidean spaces. The traditional objects of differential geometry are finite and infinite-dimensional [[differentiable manifold|differentiable manifolds]] modelled locally on [[topological vector space|topological vector spaces]]. Techniques of differential calculus can be further stretched to [[generalized smooth space|generalized smooth spaces]]. One often distinguished analysis on manifolds from differential geometry: analysis on manifolds focuses on functions from a manifold to the ground field and their properties, together with applications like PDEs on manifolds. Differential geometry on the other hand studies objects embedded into the manifold like submanifolds, their relations and additional structures on manifolds like bundles, connections etc. while the topological aspects are studied in a younger branch (from 1950s on) which is called [[differential topology]].

## Generalized smooth spaces from $n$POV

See also [[generalized smooth space]].

Finite-dimensional differential geometry is the [[geometry]] modeled on [[Cartesian space|Cartesian spaces]] and [[smooth functions]] between them.

Formally, it is the [[higher geometry|geometry]] modeled on
the [[pregeometry (for structured (infinity,1)-toposes)|pre-geometry]] $\mathcal{G} = $[[CartSp]].

This includes a sequence of concepts of [[generalized smooth spaces]]:

* A [[smooth manifold]] (see there for details) is a locally $CartSp$-representable object in the [[sheaf topos]] $Sh(CartSp)$.

* A [[diffeological space]] (see there) is a [[concrete sheaf]] in the [[cohesive topos]] $Sh(CartSp)$.

* a [[Lie groupoid]] is a locally representable object in the [[(2,1)-sheaf]] [[(2,1)-topos]] $Sh_{(2,1)}(CartSp)$;

* an [[∞-Lie groupoid]] is an object in the [[(∞,1)-sheaf (∞,1)-topos]] $Sh_{(\infty,1)}(CartSp)$.

Similarly, standard models of [[synthetic differential geometry]] in [[higher geometry]] are modeled on the [[pregeometry (for structured (infinity,1)-toposes)|pre-geometry]] $\mathcal{G} = $[[ThCartSp]]. To wit, the [[cohesive topos]] $Sh(ThCartSp)$ is the [[smooth topos]] called the [[Cahiers topos]]:

* an [[infinitesimal space]] is a certain object in $ThCartSp$;

* an [[∞-Lie algebroid]] is a certain object in the [[(∞,1)-category of (∞,1)-sheaves]] $Sh_{(\infty,1)}(ThCartSp)$.

## Related concepts

* [[trigonometry]]

* [[differential geometry of curves and surfaces]]

* [[synthetic differential geometry]]

* [[higher differential geometry]], [[derived differential geometry]]

  * [[higher differential geometry applied to plain differential geometry]]

* [[differential cohesive homotopy type theory]]

* [[differential equations]], [[geometric analysis]]

* [[prequantum geometry]], [[higher prequantum geometry]]

* [[D-geometry]]

* [[arithmetic differential geometry]]

* [[Lie theory]]

* [[fiber bundles in physics]]

* [[tangent]], [[tangent vector]], [[tangent bundle]]

* some modern subfields of differential geometry include:

  * [[symplectic geometry]],

  * [[contact geometry]],

  * [[complex geometry]],

  * [[Riemannian geometry]],

  * [[Finsler geometry]],

  * [[symmetric spaces]],

  * [[Fréchet manifolds]]

$\,$

 local model                |   global geometry
----------------------------|-----------------
 [[Klein geometry]]         |   [[Cartan geometry]]
 [[Klein 2-geometry]]       |   [[Cartan 2-geometry]]
 [[higher Klein geometry]]  |   [[higher Cartan geometry]]

$\,$

[[!include geometries of physics -- table]]

## References
 {#References}

### Diff geometry of curves and surfaces
 {#ReferencesDiffGeometryOfCurvesAndSurfaces}

The study of differential geometry goes back to the special case of [[differential geometry of curves and surfaces]]:

the study of [[curves]]  and [[surfaces]] [[embedding of smooth manifolds|embedded]] into [[Euclidean space]] $\mathbb{R}^3$:

* {#Gauss1827} [[Carl Friedrich Gauss]], _General Investigations of Curved Surfaces_ (1827) ([Gutenberg](http://www.gutenberg.org/ebooks/36856))

* Manfredo P. Do Carmo, *Differential Geometry of Curves and Surfaces*, Prentice-Hall (1976) [[pdf](http://www2.ing.unipi.it/griff/files/dC.pdf)]

* Victor A. Toponogov, *Differential Geometry of Curves and Surfaces --- A Concise Guide*, Springer (2006) [[doi:10.1007/b137116](https://doi.org/10.1007/b137116)]

* Kristopher Tapp, *Differential Geometry of Curves and Surfaces*, Springer (2016) $[$[doi:10.1007/978-3-319-39799-3](https://doi.org/10.1007/978-3-319-39799-3),  [pdf](https://link.springer.com/content/pdf/10.1007/978-3-319-39799-3.pdf)$]$

* Anton Petrunin, Sergio Zamora Barrera, *What is differential geometry: curves and surfaces* $[$[arXiv:2012.11814](https://arxiv.org/abs/2012.11814)$]$

### General but traditional diff geometry

* [[Shoshichi Kobayashi]], [[Katsumi Nomizu]], _Foundations of differential geometry_, Volume 1 (1963), Volume 2 (1969) Interscience Publishers, reprinted by Wiley Classics Library (1996) [[ISBN:978-0-470-55558-3](https://www.wiley.com/en-us/Foundations+of+Differential+Geometry%2C+2+Volume+Set-p-9780470555583), [Wikipedia entry](https://en.wikipedia.org/wiki/Foundations_of_Differential_Geometry)]

* [[Victor Guillemin]], [[Shlomo Sternberg]], *Geometric asymptotics*, Mathematical Surveys and Monographs **14**, AMS (1977) [[ams:surv-14](https://bookstore.ams.org/surv-14)]

* [[Yvonne Choquet-Bruhat]], [[Cécile DeWitt-Morette]], *Analysis, manifolds and physics*, North Holland (1982, 2001) $[$[ISBN:9780444860170](https://www.elsevier.com/books/analysis-manifolds-and-physics-revised-edition/choquet-bruhat/978-0-444-86017-0)$]$

  > (with an eye towards [[mathematical physics]])

* [[Michael Spivak]], _A comprehensive introduction to differential geometry_ (5 Volumes)

* [[Michael Spivak]], _[[Calculus on Manifolds]]_ (1971)

* [[M M Postnikov]], _Lectures on geometry_ 2001 (6 vols.: 1 "Analytic geometry", 2 "Linear algebra", 3 "Diff. manifolds"; 4 "Diff. geometry" (covers extensively fibre bundles and connections); 5 "Lie groups"; 6 "Riemannian geometry")

* [[Peter W. Michor]], _Topics in Differential Geometry_, Graduate Studies in Mathematics 93 (2008) [[pdf](https://www.mat.univie.ac.at/~michor/dgbook.pdf)]

* {#Lee12} [[John Lee]], *Introduction to Smooth Manifolds*, Springer (2012) [[doi:10.1007/978-1-4419-9982-5](https://doi.org/10.1007/978-1-4419-9982-5), [book webpage](https://sites.math.washington.edu/~lee/Books/ISM/), [pdf](https://math.berkeley.edu/~jchaidez/materials/reu/lee_smooth_manifolds.pdf)]

* [[Mikhail O. Katanaev]], *Geometrical methods in mathematical physics* (in Russian) [[arXiv1311.0733](https://arxiv.org/abs/1311.0733)]

* [[Loring Tu]], _Differential Geometry -- Connections, Curvature, and Characteristic Classes_, Springer 2017 ([ISBN:978-3-319-55082-4](https://www.springer.com/gp/book/9783319550824))

* {#GallierQuaintance20a} [[Jean Gallier]], [[Jocelyn Quaintance]],  *Differential Geometry and Lie Groups -- A computational perspective*, Geometry and Computing **12**, Springer (2020) [[doi:10.1007/978-3-030-46040-2](https://doi.org/10.1007/978-3-030-46040-2), [webpage](https://www.cis.upenn.edu/~jean/gbooks/manif.html)]

* {#GallierQuaintance20b} [[Jean Gallier]], [[Jocelyn Quaintance]],  *Differential Geometry and Lie Groups -- A second course*, Geometry and Computing **13**, Springer (2020) [[doi:10.1007/978-3-030-46047-1](https://doi.org/10.1007/978-3-030-46047-1), [webpage](https://www.cis.upenn.edu/~jean/gbooks/manif.html)]

With emphasis on [[G-structures]]:

* {#Sternberg64} [[Shlomo Sternberg]], _Lectures on differential geometry_, Prentice-Hall (1964)

With emphasis on [[natural bundles]]:

* {#KolarSlovakMichor93} [[Ivan Kolář]], [[Peter Michor]], [[Jan Slovák]], _[[Natural operations in differential geometry]]_, Springer (1993) [[book webpage](http://www.emis.de/monographs/KSM/), [doi:10.1007/978-3-662-02950-3](https://link.springer.com/book/10.1007/978-3-662-02950-3)
[pdf](http://www.emis.de/monographs/KSM/kmsbookh.pdf)]

With emphasis on [[Cartan geometry]]:

* R. Sharpe, _Differential geometry -- Cartan's generalization of Klein's Erlagen program_, Springer (1997)

Lecture notes:

* {#Conrad} [[Brian Conrad]], Handouts on Differential Geometry ([web](http://math.stanford.edu/~conrad/diffgeomPage/handouts.html))

* [[Liviu Nicolaescu]], _Lectures on the Geometry of Manifolds_, 2018 ([pdf](https://www3.nd.edu/~lnicolae/Lectures.pdf))

An introduction with an eye towards applications in [[physics]], specifically to [[gravity]] and [[gauge theory]]:

* [[Theodore Frankel]], _[[The Geometry of Physics - An Introduction]]_

* [[Chris Isham]], _[[Modern Differential Geometry for Physicists]]_

A discussion in the context of [[Frölicher spaces]] and [[diffeological spaces]] is in

* [[Andreas Kriegl]], [[Peter Michor]], _[[The convenient setting of global analysis]]_, Math. Surveys and Monographs __53__, Amer. Math. Soc. 1997. 618 pages

See also

* Sigmundur Gudmundsson, _An Introduction to Riemannian Geometry_ ([pdf](http://www.maths.lth.se/matematiklu/personal/sigma/Riemann.pdf))

* Wikipedia, _[differential geometry](http://en.wikipedia.org/wiki/Differential_geometry)_

### Higher diff geometry

See at _[[higher differential geometry]]_.

### Derived diff geometry

For [[derived differential geometry]] see

* [[Dominic Joyce]], _D-manifolds and d-orbifolds: a theory of derived differential geometry_ ([web](http://people.maths.ox.ac.uk/joyce/dmanifolds.html))

* [[Urs Schreiber]], _[[schreiber:Seminar on derived differential geometry]]_


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#Contents#
* table of contents
{:toc}

## Idea

While modern [[differential geometry]] has become a very general subject, studying [[smooth manifolds]] in full generality, the origins of the subject (before [[general relativity]] came along) lie in the study of just [[curves]] and [[surfaces]] [[embedding of smooth manifolds|embedded]] in a [[Cartesian space]]/[[Euclidean space]].

Expositions of this *differential geometry of curves and surfaces* tend to retain some of the historical ideosyncracies (such as the notion of *[[jerk]]*) which are rarely seen in more general texts on differential geometry.

## Related concepts

* [[Euclidean geometry]]

* [[curve]]

  * [[velocity]], [[areal velocity]]

  * [[acceleration]]

  * [[jerk]]

* [[surface]]

  * [[Theorema Egregium]]

(...)

## References

Historical origins:

* {#Gauss1827} [[Carl Friedrich Gauss]], _General Investigations of Curved Surfaces_ (1827) ([Gutenberg](http://www.gutenberg.org/ebooks/36856))

Modern accounts:

* Manfredo P. Do Carmo, *Differential Geometry of Curves and Surfaces*, Prentice-Hall (1976) $[$[pdf](http://www2.ing.unipi.it/griff/files/dC.pdf)$]$

* Kristopher Tapp, *Differential Geometry of Curves and Surfaces*, Springer (2016) $[$[doi:10.1007/978-3-319-39799-3](https://doi.org/10.1007/978-3-319-39799-3),  [pdf](https://link.springer.com/content/pdf/10.1007/978-3-319-39799-3.pdf)$]$

* Anton Petrunin, Sergio Zamora Barrera, *What is differential geometry: curves and surfaces* $[$[arXiv:2012.11814](https://arxiv.org/abs/2012.11814)$]$

See also:

* Wikipedia, *[Differential geometry of curves](https://en.wikipedia.org/wiki/Differential_geometry_of_curves)*

* Wikipedia, *[Differential geometry of surfaces](https://en.wikipedia.org/wiki/Differential_geometry_of_surfaces)*

[[!redirects differential geometry of curves]]
[[!redirects differential geometry of surfaces]]