Differentiable manifold

See also: Diffeology

In mathematics, a **differentiable manifold** (also **differential manifold**) is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since each chart lies within a linear space to which the usual rules of calculus apply. If the charts are suitably compatible (namely, the transition from one chart to another is differentiable), then computations done in one chart are valid in any other differentiable chart.

In formal terms, a **differentiable manifold** is a topological manifold with a globally defined differential structure. Any topological manifold can be given a differential structure *locally* by using the homeomorphisms in its atlas and the standard differential structure on a linear space. To induce a global differential structure on the local coordinate systems induced by the homeomorphisms, their composition on chart intersections in the atlas must be differentiable functions on the corresponding linear space. In other words, where the domains of charts overlap, the coordinates defined by each chart are required to be differentiable with respect to the coordinates defined by every chart in the atlas. The maps that relate the coordinates defined by the various charts to one another are called *transition maps.*

Differentiability means different things in different contexts including: continuously differentiable, *k* times differentiable, smooth, and holomorphic. Furthermore, the ability to induce such a differential structure on an abstract space allows one to extend the definition of differentiability to spaces without global coordinate systems. A differential structure allows one to define the globally differentiable tangent space, differentiable functions, and differentiable tensor and vector fields. Differentiable manifolds are very important in physics. Special kinds of differentiable manifolds form the basis for physical theories such as classical mechanics, general relativity, and Yang–Mills theory. It is possible to develop a calculus for differentiable manifolds. This leads to such mathematical machinery as the exterior calculus. The study of calculus on differentiable manifolds is known as differential geometry.

## History

Main articles: History of manifolds and varieties
The emergence of differential geometry as a distinct discipline is generally credited to Carl Friedrich Gauss and Bernhard Riemann. Riemann first described manifolds in his famous habilitation lecture before the faculty at Göttingen. He motivated the idea of a manifold by an intuitive process of varying a given object in a new direction, and presciently described the role of coordinate systems and charts in subsequent formal developments:

*Having constructed the notion of a manifoldness of n dimensions, and found that its true character consists in the property that the determination of position in it may be reduced to n determinations of magnitude, ...* – B. Riemann
The works of physicists such as James Clerk Maxwell, and mathematicians Gregorio Ricci-Curbastro and Tullio Levi-Civita led to the development of tensor analysis and the notion of covariance, which identifies an intrinsic geometric property as one that is invariant with respect to coordinate transformations. These ideas found a key application in Albert Einstein's theory of general relativity and its underlying equivalence principle. A modern definition of a 2-dimensional manifold was given by Hermann Weyl in his 1913 book on Riemann surfaces. The widely accepted general definition of a manifold in terms of an atlas is due to Hassler Whitney.

## Definition

A topological n-manifold is a topological space

M

{\displaystyle M}

 which satisfies the following three conditions:

* it is Hausdorff
* it is second countable
* each element of

M

{\displaystyle M}

 is contained in an open set which is homeomorphic to an open set in

R

n

.

{\displaystyle \mathbb{R}^n.}

So a topological manifold is a certain kind of topological space; if one is presented with a topological space, it is meaningful to ask whether or not it is a topological manifold. By contrast, the various kinds of differentiable manifolds, as defined below, are not "certain kinds" of topological manifold; instead, they endow a given topological manifold with an additional structure. This "additional structure" is the specification of certain distinguished charts, as defined below.

A "chart" on a topological space

M

{\displaystyle M}

 is an open subset

U
⊂
M
,

{\displaystyle U\subset M,}

 together with a homeomorphism from

U

{\displaystyle U}

 to an open subset of

R

n

;

{\displaystyle \mathbb{R}^n;}

 a collection of charts whose domains cover

M

{\displaystyle M}

 is called an atlas. The third condition in the above definition of a topological manifold could be phrased in this terminology as saying "an atlas exists".

The composition of one chart with the inverse of another chart is a function called a 'transition map', and is automatically a continuous map (and even a homeomorphism) from one open subset of

R

n

{\displaystyle \R^n}

 to another.

The following defines some common types of atlases:

* An atlas is called a ***C**k*-atlas** if all of its transition maps are *k*-times continuously differentiable.
* An atlas is called a **smooth or *C*∞-atlas** if the transition maps are all smooth, i.e. all derivatives of all orders exist (and are then necessarily continuous).
* An atlas is called an **analytic or *C*ω-atlas** if all of the transition maps are analytic, i.e. the Taylor series around an arbitrary point converges to the function on some open ball.
* An atlas on a topological 2n-manifold is called a **holomorphic or complex atlas** if, after identifying

R

2
n

{\displaystyle \mathbb R^{2n}}

 with

C

n

{\displaystyle \C^n}

 via

(

x
1

,
…
,

x

2
n

)
↔
(

x
1

−
1

x

n
1

,
…
,

x
n

−
1

x

2
n

)
,

{\displaystyle (x^1,\ldots,x^{2n})\leftrightarrow (x^1+\sqrt{-1}x^{n+1},\ldots,x^n+\sqrt{-1}x^{2n}),}

all transition maps are holomorphic.
In the following four paragraphs, for convenience we will consider only smooth atlases. Everything would remain true if one replaced the word "smooth" everywhere it appears with either "holomorphic," "analytic," or "*C**k*."

If one is given a smooth atlas and some other chart on the topological manifold, one says that the chart is smooth (implicitly speaking relative to the given atlas) if its transition maps with each element of the atlas are smooth. Alternatively, one would say that the chart is "compatible" with the atlas.

In practice, if one is explicitly dealing with atlases, the number of charts involved is often quite small - often even as few as two or three. However, for abstract purposes, it can be useful to consider the "maximal" smooth atlas defined by a given smooth atlas, which consists of the given atlas together with all charts which are compatible with it.

A smooth *n*-manifold is a topological *n*-manifold together with a maximal smooth atlas.

A maximal atlas is always enormous; for instance, if it contains a chart

φ
:
U
→

R

n

,

{\displaystyle \varphi:U\to\mathbb{R}^n,}

 then it necessarily also contains the restriction of

φ

{\displaystyle \varphi}

 to every open subset of

U
.

{\displaystyle U.}

 An alternative definition, which can be more appropriate in certain practical circumstances, is to say that a smooth *n*-manifold is a topological *n*-manifold together with an equivalence class of smooth atlases, in which two smooth atlases are considered equivalent if every chart of one atlas is compatible with the other atlas. Informally, what this means is that in dealing with a smooth manifold, one can work with a topological manifold together with a smooth atlas consisting of only a few charts, with the implicit understanding that many other charts are equally legitimate.

### Atlases

M

{\displaystyle M}

U
α

{\displaystyle U\_\alpha}

U
β

{\displaystyle U\_\beta}

φ
α

{\displaystyle \varphi\_\alpha}

φ
β

{\displaystyle \varphi\_\beta}

φ

α
β

{\displaystyle \varphi\_{\alpha\beta}}

φ

β
α

{\displaystyle \varphi\_{\beta\alpha}}

R

n

{\displaystyle \mathbf R^n}

R

n

{\displaystyle \mathbf R^n}

Here we provide common notation and the formal definitions which appear in the above discussion. We will use the smooth context for convenience; one could replace the word "smooth" everywhere it appears with one of "*C**k*," "analytic," or "holomorphic."

Let *M* be a topological *n*-manifold. A "chart" on *M* consists of an open subset *U* of *M* and a homeomorphism φ from *U* to an open subset of *n*-dimensional Euclidean space. An "atlas" on *M* consists of an indexing set *A* and a collection of charts

{
(

U
α

,

φ
α

)

}

α
∈
A

{\displaystyle \{(U\_\alpha,\varphi\_\alpha)\}\_{\alpha\in A}}

 such that the union of the domains *U**α* is *M*. (One says that {*U**α*} "cover" *M*.)

The **transition maps** of the atlas are the functions

φ

α
β

≡

φ
β

∘

φ
α

−
1

φ
α

(

U
α

∩

U
β

)

:

φ
α

(

U
α

∩

U
β

)
→

φ
β

(

U
α

∩

U
β

)

{\displaystyle \varphi\_{\alpha\beta} \equiv \varphi\_\beta\circ\varphi\_\alpha^{-1}|\_{\varphi\_\alpha(U\_\alpha\cap U\_\beta)} \colon \varphi\_\alpha(U\_\alpha\cap U\_\beta) \to \varphi\_\beta(U\_\alpha\cap U\_\beta)}

for every choice

α
,
β
∈
A
,

{\displaystyle \alpha,\beta\in A,}

 provided only that

U
α

∩

U
β

{\displaystyle U\_\alpha\cap U\_\beta}

 is nonempty.

One says that an atlas is smooth if all of its transition maps are smooth. For the remainder of this section, we will consider

One says that two smooth atlases on the same topological manifold are **compatible** if their union remains a smooth atlas. It is straightforward to check that this defines an equivalence relation on the collection of all smooth atlases.

A smooth *n*-manifold consists of a topological *n*-manifold together with an equivalence class of smooth atlases. In particular, a given smooth atlas on a topological *n*-manifold *defines* a smooth *n*-manifold.

For an alternative definition, one says that a smooth atlas is **maximal** if it contains as a subset every smooth atlas that it is compatible with. A smooth *n*-manifold can then be considered as a topological *n*-manifold together with a maximal smooth atlas.

### Analytic manifolds as smooth manifolds

Since any holomorphic function is analytic, and any analytic function is smooth, and any smooth function is

C
k

{\displaystyle C^k}

 for any

k
∈

N

,

{\displaystyle k \in \mathbb{N},}

 one can immediately conclude that any holomorphic atlas can be considered, in a natural way, as an analytic atlas; any analytic atlas can be considered as a smooth atlas, and any smooth atlas can be considered as a *C**k* atlas for any *k*.

Taking this only one step further, any complex *n*-manifold can be considered, naturally, as an analytic 2*n*-manifold; an analytic *n*-manifold can be considered as a smooth *n*-manifold, and a smooth *n*-manifold can be considered as a *C**k* *n*-manifold for any *k*.

However, since there exist (for instance) smooth functions between open subsets of

R

2
n

{\displaystyle \mathbb R^{2n}}

 which are not holomorphic when considered as mappings between open subsets of

C

n

,

{\displaystyle \mathbb{C}^n,}

 the converse is not automatically true. There are some subtleties in addressing this properly; the issue is discussed at greater length in the article on differential structure.

### Patching together Euclidean pieces to form a manifold

One can reverse-engineer the above definitions to obtain one perspective on the construction of manifolds. The idea is to start with the images of the charts and the transition maps, and to construct the manifold purely from this data. As in the above discussion, we use the "smooth" context but everything works just as well in other settings.

Given an indexing set

A
,

{\displaystyle A,}

 let

V
α

{\displaystyle V\_\alpha}

 be a collection of open subsets of

R

n

{\displaystyle \R^n}

 and for each

α
,
β
∈
A

{\displaystyle \alpha, \beta \in A}

 let

V

α
β

{\displaystyle V\_{\alpha\beta}}

 be an open (possibly empty) subset of

V
β

{\displaystyle V\_\beta}

 and let

φ

α
β

:

V

α
β

→

V

β
α

{\displaystyle \varphi\_{\alpha\beta}:V\_{\alpha\beta}\to V\_{\beta\alpha}}

 be a smooth map. Suppose that

φ

α
α

{\displaystyle \varphi\_{\alpha\alpha}}

 is the identity map, that

φ

α
β

∘

φ

β
α

{\displaystyle \varphi\_{\alpha\beta}\circ\varphi\_{\beta\alpha}}

 is the identity map, and that

φ

α
β

∘

φ

β
γ

∘

φ

γ
α

{\displaystyle \varphi\_{\alpha\beta}\circ\varphi\_{\beta\gamma}\circ\varphi\_{\gamma\alpha}}

 is the identity map. Then define an equivalence relation on the disjoint union

⨆

α
∈
A

V
α

{\displaystyle \bigsqcup\_{\alpha\in A}V\_\alpha}

 by declaring

p
∈

V

α
β

{\displaystyle p\in V\_{\alpha\beta}}

 to be equivalent to

φ

α
β

(
p
)
∈

V

β
α

.

{\displaystyle \varphi\_{\alpha\beta}(p)\in V\_{\beta\alpha}.}

 With some technical work, one can show that the set of equivalence classes can naturally be given a topological structure, and that the charts used in doing so form a smooth atlas.

## Alternative definitions

### Pseudogroups

The notion of a pseudogroup provides a flexible generalization of atlases in order to allow a variety of different structures to be defined on manifolds in a uniform way. A *pseudogroup* consists of a topological space *S* and a collection Γ consisting of homeomorphisms from open subsets of *S* to other open subsets of *S* such that

1. If *f* ∈ Γ, and *U* is an open subset of the domain of *f*, then the restriction *f*|*U* is also in Γ.
2. If *f* is a homeomorphism from a union of open subsets of *S*,

∪
i

U
i

{\displaystyle \cup\_i \, U\_i }

, to an open subset of *S*, then *f* ∈ Γ provided

f

U
i

∈
Γ

{\displaystyle f|\_{U\_i} \in \Gamma }

 for every *i*.
3. For every open *U* ⊂ *S*, the identity transformation of *U* is in Γ.
4. If *f* ∈ Γ, then *f*−1 ∈ Γ.
5. The composition of two elements of Γ is in Γ.

These last three conditions are analogous to the definition of a group. Note that Γ need not be a group, however, since the functions are not globally defined on *S*. For example, the collection of all local *Ck* diffeomorphisms on **R***n* form a pseudogroup. All biholomorphisms between open sets in **C***n* form a pseudogroup. More examples include: orientation preserving maps of **R***n*, symplectomorphisms, Möbius transformations, affine transformations, and so on. Thus, a wide variety of function classes determine pseudogroups.

An atlas (*Ui*, *φ**i*) of homeomorphisms *φ**i* from *Ui* ⊂ *M* to open subsets of a topological space *S* is said to be *compatible* with a pseudogroup Γ provided that the transition functions *φ**j* ∘ *φ**i*−1 : *φ**i*(*Ui* ∩ *Uj*) → *φ**j*(*Ui* ∩ *Uj*) are all in Γ.

A differentiable manifold is then an atlas compatible with the pseudogroup of *C**k* functions on **R***n*. A complex manifold is an atlas compatible with the biholomorphic functions on open sets in **C***n*. And so forth. Thus, pseudogroups provide a single framework in which to describe many structures on manifolds of importance to differential geometry and topology.

### Structure sheaf

Sometimes, it can be useful to use an alternative approach to endow a manifold with a *Ck*-structure. Here *k* = 1, 2, ..., ∞, or ω for real analytic manifolds. Instead of considering coordinate charts, it is possible to start with functions defined on the manifold itself. The structure sheaf of *M*, denoted **C***k*, is a sort of functor that defines, for each open set *U* ⊂ *M*, an algebra **C***k*(*U*) of continuous functions *U* → **R**. A structure sheaf **C***k* is said to give *M* the structure of a *C**k* manifold of dimension *n* provided that, for any *p* ∈ *M*, there exists a neighborhood *U* of *p* and *n* functions *x*1, ..., *x**n* ∈ **C***k*(*U*) such that the map *f* = (*x*1, ..., *xn*) : *U* → **R***n* is a homeomorphism onto an open set in **R***n*, and such that **C***k*|*U* is the pullback of the sheaf of *k*-times continuously differentiable functions on **R***n*.

In particular, this latter condition means that any function *h* in **C***k*(*V*), for *V*, can be written uniquely as *h*(*x*) = *H*(*x*1(*x*), ..., *x**n*(*x*)), where *H* is a *k*-times differentiable function on *f*(*V*) (an open set in **R***n*). Thus, the sheaf-theoretic viewpoint is that the functions on a differentiable manifold can be expressed in local coordinates as differentiable functions on **R***n*, and *a fortiori* this is sufficient to characterize the differential structure on the manifold.

#### Sheaves of local rings

A similar, but more technical, approach to defining differentiable manifolds can be formulated using the notion of a ringed space. This approach is strongly influenced by the theory of schemes in algebraic geometry, but uses local rings of the germs of differentiable functions. It is especially popular in the context of *complex* manifolds.

We begin by describing the basic structure sheaf on **R***n*. If *U* is an open set in **R***n*, let

**O**(*U*) = *C**k*(*U*, **R**)
consist of all real-valued *k*-times continuously differentiable functions on *U*. As *U* varies, this determines a sheaf of rings on **R**n. The stalk **O***p* for *p* ∈ **R***n* consists of germs of functions near *p*, and is an algebra over **R**. In particular, this is a local ring whose unique maximal ideal consists of those functions that vanish at *p*. The pair (**R***n*, **O**) is an example of a locally ringed space: it is a topological space equipped with a sheaf whose stalks are each local rings.

A differentiable manifold (of class *Ck*) consists of a pair (*M*, **O***M*) where *M* is a second countable Hausdorff space, and **O***M* is a sheaf of local **R**-algebras defined on *M*, such that the locally ringed space (*M*, **O***M*) is locally isomorphic to (**R***n*, **O**). In this way, differentiable manifolds can be thought of as schemes modelled on **R***n*. This means that for each point *p* ∈ *M*, there is a neighborhood *U* of *p*, and a pair of functions (*f*, *f*#), where

1. *f* : *U* → *f*(*U*) ⊂ **R***n* is a homeomorphism onto an open set in **R***n*.
2. *f*#: **O**|*f*(*U*) → *f*∗ (**O***M*|*U*) is an isomorphism of sheaves.
3. The localization of *f*# is an isomorphism of local rings

*f*#*f*(*p*) : **O***f*(*p*) → **O***M*,*p*.
There are a number of important motivations for studying differentiable manifolds within this abstract framework. First, there is no *a priori* reason that the model space needs to be **R**n. For example, (in particular in algebraic geometry), one could take this to be the space of complex numbers **C***n* equipped with the sheaf of holomorphic functions (thus arriving at the spaces of complex analytic geometry), or the sheaf of polynomials (thus arriving at the spaces of interest in complex *algebraic* geometry). In broader terms, this concept can be adapted for any suitable notion of a scheme (see topos theory). Second, coordinates are no longer explicitly necessary to the construction. The analog of a coordinate system is the pair (*f*, *f*#), but these merely quantify the idea of *local isomorphism* rather than being central to the discussion (as in the case of charts and atlases). Third, the sheaf **O***M* is not manifestly a sheaf of functions at all. Rather, it emerges as a sheaf of functions as a *consequence* of the construction (via the quotients of local rings by their maximal ideals). Hence, it is a more primitive definition of the structure (see synthetic differential geometry).

A final advantage of this approach is that it allows for natural direct descriptions of many of the fundamental objects of study to differential geometry and topology.

* The cotangent space at a point is *Ip*/*Ip*2, where *Ip* is the maximal ideal of the stalk **O***M*,*p*.
* In general, the entire cotangent bundle can be obtained by a related technique (see cotangent bundle for details).
* Taylor series (and jets) can be approached in a coordinate-independent manner using the *I**p*-adic filtration on **O***M*,*p*.
* The tangent bundle (or more precisely its sheaf of sections) can be identified with the sheaf of morphisms of **O***M* into the ring of dual numbers.

## Differentiable functions

A real valued function *f* on an *n*-dimensional differentiable manifold *M* is called **differentiable** at a point *p* ∈ *M* if it is differentiable in any coordinate chart defined around *p*. In more precise terms, if

(
U
,
φ
)

{\displaystyle (U,\varphi)}

 is a differentiable chart where

U

{\displaystyle U}

 is an open set in

M

{\displaystyle M}

 containing *p* and

φ
:
U
→

R

n

{\displaystyle \varphi : U\to {\mathbf R}^n}

 is the map defining the chart, then *f* is differentiable at *p* if and only if

f
∘

φ

−
1

:
φ
(
U
)
⊂

R

n

→

R

{\displaystyle f\circ \varphi^{-1} \colon \varphi(U)\subset {\mathbf R}^n \to {\mathbf R}}

is differentiable at

φ
(
p
)

{\displaystyle \varphi(p)}

, that is *f* is a differentiable function from the open set

φ
(
U
)

{\displaystyle \varphi(U)}

, considered as a subset of

R

n

{\displaystyle {\mathbf R}^n}

, to

R

{\displaystyle \mathbf{R}}

. In general, there will be many available charts; however, the definition of differentiability does not depend on the choice of chart at *p*. It follows from the chain rule applied to the transition functions between one chart and another that if *f* is differentiable in any particular chart at *p*, then it is differentiable in all charts at *p*. Analogous considerations apply to defining *Ck* functions, smooth functions, and analytic functions.

### Differentiation of functions

There are various ways to define the derivative of a function on a differentiable manifold, the most fundamental of which is the directional derivative. The definition of the directional derivative is complicated by the fact that a manifold will lack a suitable affine structure with which to define vectors. Therefore, the directional derivative looks at curves in the manifold instead of vectors.

#### Directional differentiation

Given a real valued function *f* on an *n* dimensional differentiable manifold *M*, the directional derivative of *f* at a point *p* in *M* is defined as follows. Suppose that γ(*t*) is a curve in *M* with *γ*(0) = *p*, which is *differentiable* in the sense that its composition with any chart is a differentiable curve in **R***n*. Then the **directional derivative** of *f* at *p* along γ is

d

d
t

f
(
γ
(
t
)
)

# t
0

.

{\displaystyle \left.\frac{d}{dt}f(\gamma(t))\right|\_{t=0}.}

If *γ*1 and *γ*2 are two curves such that *γ*1(0) = *γ*2(0) = *p*, and in any coordinate chart *φ*,

d

d
t

φ
∘

γ
1

(
t
)

# t
0

=

d

d
t

φ
∘

γ
2

(
t
)

# t
0

{\displaystyle \left.\frac{d}{dt}\varphi\circ\gamma\_1(t)\right|\_{t=0}=\left.\frac{d}{dt}\varphi\circ\gamma\_2(t)\right|\_{t=0}}

then, by the chain rule, *f* has the same directional derivative at *p* along *γ*1 as along *γ*2. This means that the directional derivative depends only on the tangent vector of the curve at *p*. Thus, the more abstract definition of directional differentiation adapted to the case of differentiable manifolds ultimately captures the intuitive features of directional differentiation in an affine space.

#### Tangent vector and the differential

A **tangent vector** at *p* ∈ *M* is an equivalence class of differentiable curves *γ* with *γ*(0) = *p*, modulo the equivalence relation of first-order contact between the curves. Therefore,

γ
1

≡

γ
2

⟺

d

d
t

φ
∘

γ
1

(
t
)

# t
0

=

d

d
t

φ
∘

γ
2

(
t
)

# t
0

{\displaystyle \gamma\_1\equiv \gamma\_2 \iff \left.\frac{d}{dt}\varphi\circ\gamma\_1(t)\right|\_{t=0} = \left.\frac{d}{dt}\varphi\circ\gamma\_2(t)\right|\_{t=0}}

in every coordinate chart *φ*. Therefore, the equivalence classes are curves through *p* with a prescribed velocity vector at *p*. The collection of all tangent vectors at *p* forms a vector space: the tangent space to *M* at *p*, denoted *T**p**M*.

If *X* is a tangent vector at *p* and *f* a differentiable function defined near *p*, then differentiating *f* along any curve in the equivalence class defining *X* gives a well-defined directional derivative along *X*:

X
f
(
p
)
:=

d

d
t

f
(
γ
(
t
)
)

# t
0

.

{\displaystyle Xf(p) := \left.\frac{d}{dt}f(\gamma(t))\right|\_{t=0}.}

Once again, the chain rule establishes that this is independent of the freedom in selecting γ from the equivalence class, since any curve with the same first order contact will yield the same directional derivative.

If the function *f* is fixed, then the mapping

X
↦
X
f
(
p
)

{\displaystyle X\mapsto Xf(p)}

is a linear functional on the tangent space. This linear functional is often denoted by *df*(*p*) and is called the **differential** of *f* at *p*:

d
f
(
p
)
:

T
p

M
→

R

.

{\displaystyle df(p) \colon T\_pM \to {\mathbf R}.}

#### Definition of tangent space and differentiation in local coordinates

Let

M

{\displaystyle M}

 be a topological

n

{\displaystyle n}

-manifold with a smooth atlas

{
(

U
α

,

φ
α

)

}

α
∈
A

.

{\displaystyle \{(U\_\alpha,\varphi\_\alpha)\}\_{\alpha\in A}.}

 Given

p
∈
M

{\displaystyle p \in M}

 let

A

p

{\displaystyle A\_{p}}

 denote

{
α
∈
A
:
p
∈

U
α

}
.

{\displaystyle \{\alpha\in A:p\in U\_\alpha\}.}

 A "tangent vector at

p
∈
M

{\displaystyle p \in M}

" is a mapping

v
:

A
p

→

R

n

,

{\displaystyle v:A\_p\to\mathbb{R}^n,}

 here denoted

α
↦

v
α

,

{\displaystyle \alpha\mapsto v\_\alpha,}

 such that

v
α

=
D

φ
β

(
p
)

(

φ
α

∘

φ
β

−
1

)
(

v
β

)

{\displaystyle v\_\alpha=D\Big|\_{\varphi\_\beta(p)}(\varphi\_\alpha\circ\varphi\_\beta^{-1})(v\_\beta)}

for all

α
,
β
∈

A
p

.

{\displaystyle \alpha,\beta\in A\_p.}

 Let the collection of tangent vectors at

p

{\displaystyle p}

 be denoted by

T
p

M
.

{\displaystyle T\_pM.}

 Given a smooth function

f
:
M
→

R

{\displaystyle f:M\to\mathbb{R}}

, define

d

f
p

:

T
p

M
→

R

{\displaystyle df\_p:T\_pM\to\mathbb{R}}

 by sending a tangent vector

v
:

A
p

→

R

n

{\displaystyle v:A\_p\to\mathbb{R}^n}

 to the number given by

D

φ
α

(
p
)

(
f
∘

φ
α

−
1

)
(

v
α

)
,

{\displaystyle D\Big|\_{\varphi\_\alpha(p)}(f\circ\varphi\_\alpha^{-1})(v\_\alpha),}

which due to the chain rule and the constraint in the definition of a tangent vector does not depend on the choice of

α
∈

A
p

.

{\displaystyle \alpha\in A\_p.}

One can check that

T
p

M

{\displaystyle T\_p M}

 naturally has the structure of a

n

{\displaystyle n}

-dimensional real vector space, and that with this structure,

d

f
p

{\displaystyle df\_p}

 is a linear map. The key observation is that, due to the constraint appearing in the definition of a tangent vector, the value of

v
β

{\displaystyle v\_\beta}

 for a single element

β

{\displaystyle \beta}

 of

A

p

{\displaystyle A\_{p}}

 automatically determines

v
α

{\displaystyle v\_\alpha}

 for all

α
∈
A
.

{\displaystyle \alpha\in A.}

The above formal definitions correspond precisely to a more informal notation which appears often in textbooks, specifically

v
i

=

v
~

j

∂

x
i

∂

x
~

j

{\displaystyle v^i=\widetilde{v}^j\frac{\partial x^i}{\partial\widetilde{x}^j}}

 and

d

f
p

(
v
# )

∂
f

∂

x
i

v
i

.

{\displaystyle df\_p(v)=\frac{\partial f}{\partial x^i}v^i.}

With the idea of the formal definitions understood, this shorthand notation is, for most purposes, much easier to work with.

### Partitions of unity

One of the topological features of the sheaf of differentiable functions on a differentiable manifold is that it admits partitions of unity. This distinguishes the differential structure on a manifold from stronger structures (such as analytic and holomorphic structures) that in general fail to have partitions of unity.

Suppose that *M* is a manifold of class *Ck*, where 0 ≤ *k* ≤ ∞. Let {*U**α*} be an open covering of *M*. Then a **partition of unity** subordinate to the cover {*U**α*} is a collection of real-valued *Ck* functions *φ**i* on *M* satisfying the following conditions:

* The supports of the *φ**i* are compact and locally finite;
* The support of *φ**i* is completely contained in *U**α* for some *α*;
* The *φ**i* sum to one at each point of *M*:

∑
i

φ
i

(
x
# )

{\displaystyle \sum\_i \varphi\_i(x) = 1.}

(Note that this last condition is actually a finite sum at each point because of the local finiteness of the supports of the *φ**i*.)

Every open covering of a *Ck* manifold *M* has a *Ck* partition of unity. This allows for certain constructions from the topology of *Ck* functions on **R***n* to be carried over to the category of differentiable manifolds. In particular, it is possible to discuss integration by choosing a partition of unity subordinate to a particular coordinate atlas, and carrying out the integration in each chart of **R***n*. Partitions of unity therefore allow for certain other kinds of function spaces to be considered: for instance L*p* spaces, Sobolev spaces, and other kinds of spaces that require integration.

### Differentiability of mappings between manifolds

Suppose *M* and *N* are two differentiable manifolds with dimensions *m* and *n*, respectively, and *f* is a function from *M* to *N*. Since differentiable manifolds are topological spaces we know what it means for *f* to be continuous. But what does "*f* is *Ck*(*M*, *N*)" mean for *k* ≥ 1? We know what that means when *f* is a function between Euclidean spaces, so if we compose *f* with a chart of *M* and a chart of *N* such that we get a map that goes from Euclidean space to *M* to *N* to Euclidean space we know what it means for that map to be *Ck*(**R***m*, **R***n*). We define "*f* is *Ck*(*M*, *N*)" to mean that all such compositions of *f* with charts are *Ck*(**R***m*, **R***n*). Once again, the chain rule guarantees that the idea of differentiability does not depend on which charts of the atlases on *M* and *N* are selected. However, defining the derivative itself is more subtle. If *M* or *N* is itself already a Euclidean space, then we don't need a chart to map it to one.

## Bundles

### Tangent bundle

Further information: tangent bundle
The tangent space of a point consists of the possible directional derivatives at that point, and has the same dimension *n* as does the manifold. For a set of (non-singular) coordinates *xk* local to the point, the coordinate derivatives

∂
k

=

∂

∂

x
k

{\displaystyle \partial\_k=\frac{\partial}{\partial x\_k}}

 define a holonomic basis of the tangent space. The collection of tangent spaces at all points can in turn be made into a manifold, the tangent bundle, whose dimension is 2*n*. The tangent bundle is where tangent vectors lie, and is itself a differentiable manifold. The Lagrangian is a function on the tangent bundle. One can also define the tangent bundle as the bundle of 1-jets from **R** (the real line) to *M*.

One may construct an atlas for the tangent bundle consisting of charts based on *U**α* × **R***n*, where *U**α* denotes one of the charts in the atlas for *M*. Each of these new charts is the tangent bundle for the charts *U**α*. The transition maps on this atlas are defined from the transition maps on the original manifold, and retain the original differentiability class.

### Cotangent bundle

Further information: cotangent bundle
The dual space of a vector space is the set of real valued linear functions on the vector space. The cotangent space at a point is the dual of the tangent space at that point, and the cotangent bundle is the collection of all cotangent spaces.

Like the tangent bundle, the cotangent bundle is again a differentiable manifold. The Hamiltonian is a scalar on the cotangent bundle. The total space of a cotangent bundle has the structure of a symplectic manifold. Cotangent vectors are sometimes called *covectors*. One can also define the cotangent bundle as the bundle of 1-jets of functions from *M* to **R**.

Elements of the cotangent space can be thought of as infinitesimal displacements: if *f* is a differentiable function we can define at each point *p* a cotangent vector *dfp*, which sends a tangent vector *Xp* to the derivative of *f* associated with *Xp*. However, not every covector field can be expressed this way. Those that can are referred to as exact differentials. For a given set of local coordinates *xk,* the differentials *dx**k*
*p* form a basis of the cotangent space at *p*.

### Tensor bundle

Further information: tensor bundle
The tensor bundle is the direct sum of all tensor products of the tangent bundle and the cotangent bundle. Each element of the bundle is a tensor field, which can act as a multilinear operator on vector fields, or on other tensor fields.

The tensor bundle is not a differentiable manifold in the traditional sense, since it is infinite dimensional. It is however an algebra over the ring of scalar functions. Each tensor is characterized by its ranks, which indicate how many tangent and cotangent factors it has. Sometimes these ranks are referred to as *covariant* and *contravariant* ranks, signifying tangent and cotangent ranks, respectively.

### Frame bundle

Further information: frame bundle
A frame (or, in more precise terms, a tangent frame), is an ordered basis of particular tangent space. Likewise, a tangent frame is a linear isomorphism of **R***n* to this tangent space. A moving tangent frame is an ordered list of vector fields that give a basis at every point of their domain. One may also regard a moving frame as a section of the frame bundle F(*M*), a GL(*n*, **R**) principal bundle made up of the set of all frames over *M*. The frame bundle is useful because tensor fields on *M* can be regarded as equivariant vector-valued functions on F(*M*).

### Jet bundles

Further information: jet bundle
On a manifold that is sufficiently smooth, various kinds of jet bundles can also be considered. The (first-order) tangent bundle of a manifold is the collection of curves in the manifold modulo the equivalence relation of first-order contact. By analogy, the *k*-th order tangent bundle is the collection of curves modulo the relation of *k*-th order contact. Likewise, the cotangent bundle is the bundle of 1-jets of functions on the manifold: the *k*-jet bundle is the bundle of their *k*-jets. These and other examples of the general idea of jet bundles play a significant role in the study of differential operators on manifolds.

The notion of a frame also generalizes to the case of higher-order jets. Define a *k*-th order frame to be the *k*-jet of a diffeomorphism from **R***n* to *M*. The collection of all *k*-th order frames, *Fk*(*M*), is a principal *Gk* bundle over *M*, where *Gk* is the group of *k*-jets; i.e., the group made up of *k*-jets of diffeomorphisms of **R***n* that fix the origin. Note that GL(*n*, **R**) is naturally isomorphic to *G*1, and a subgroup of every *Gk*, *k* ≥ 2. In particular, a section of *F*2(*M*) gives the frame components of a connection on *M*. Thus, the quotient bundle *F*2(*M*) / GL(*n*, **R**) is the bundle of *symmetric* linear connections over *M*.

## Calculus on manifolds

Many of the techniques from multivariate calculus also apply, *mutatis mutandis*, to differentiable manifolds. One can define the directional derivative of a differentiable function along a tangent vector to the manifold, for instance, and this leads to a means of generalizing the total derivative of a function: the differential. From the perspective of calculus, the derivative of a function on a manifold behaves in much the same way as the ordinary derivative of a function defined on a Euclidean space, at least locally. For example, there are versions of the implicit and inverse function theorems for such functions.

There are, however, important differences in the calculus of vector fields (and tensor fields in general). In brief, the directional derivative of a vector field is not well-defined, or at least not defined in a straightforward manner. Several generalizations of the derivative of a vector field (or tensor field) do exist, and capture certain formal features of differentiation in Euclidean spaces. The chief among these are:

* The Lie derivative, which is uniquely defined by the differential structure, but fails to satisfy some of the usual features of directional differentiation.
* An affine connection, which is not uniquely defined, but generalizes in a more complete manner the features of ordinary directional differentiation. Because an affine connection is not unique, it is an additional piece of data that must be specified on the manifold.

Ideas from integral calculus also carry over to differential manifolds. These are naturally expressed in the language of exterior calculus and differential forms. The fundamental theorems of integral calculus in several variables—namely Green's theorem, the divergence theorem, and Stokes' theorem—generalize to a theorem (also called Stokes' theorem) relating the exterior derivative and integration over submanifolds.

### Differential calculus of functions

Differentiable functions between two manifolds are needed in order to formulate suitable notions of submanifolds, and other related concepts. If *f* : *M* → *N* is a differentiable function from a differentiable manifold *M* of dimension *m* to another differentiable manifold *N* of dimension *n*, then the differential of *f* is a mapping *df* : T*M* → T*N*. It is also denoted by *Tf* and called the **tangent map**. At each point of *M*, this is a linear transformation from one tangent space to another:

d
f
(
p
)
:

T
p

M
→

T

f
(
p
)

N
.

{\displaystyle df(p)\colon T\_p M \to T\_{f(p)} N.}

The **rank** of *f* at *p* is the rank of this linear transformation.

Usually the rank of a function is a pointwise property. However, if the function has maximal rank, then the rank will remain constant in a neighborhood of a point. A differentiable function "usually" has maximal rank, in a precise sense given by Sard's theorem. Functions of maximal rank at a point are called immersions and submersions:

* If *m* ≤ *n*, and *f* : *M* → *N* has rank *m* at *p* ∈ *M*, then *f* is called an **immersion** at *p*. If *f* is an immersion at all points of *M* and is a homeomorphism onto its image, then *f* is an **embedding**. Embeddings formalize the notion of *M* being a submanifold of *N*. In general, an embedding is an immersion without self-intersections and other sorts of non-local topological irregularities.
* If *m* ≥ *n*, and *f* : *M* → *N* has rank *n* at *p* ∈ *M*, then *f* is called a **submersion** at *p*. The implicit function theorem states that if *f* is a submersion at *p*, then *M* is locally a product of *N* and **R***m*−*n* near *p*. In formal terms, there exist coordinates (*y*1, ..., *yn*) in a neighborhood of *f*(*p*) in *N*, and *m* − *n* functions *x*1, ..., *x**m*−*n* defined in a neighborhood of *p* in *M* such that

(

y
1

∘
f
,
…
,

y
n

∘
f
,

x
1

,
…
,

x

m
−
n

)

{\displaystyle (y\_1\circ f,\dotsc,y\_n\circ f, x\_1, \dotsc, x\_{m-n})}

is a system of local coordinates of *M* in a neighborhood of *p*. Submersions form the foundation of the theory of fibrations and fibre bundles.
### Lie derivative

A Lie derivative, named after Sophus Lie, is a derivation on the algebra of tensor fields over a manifold *M*. The vector space of all Lie derivatives on *M* forms an infinite dimensional Lie algebra with respect to the Lie bracket defined by

[
A
,
B
]
:=

L

A

# B
−

L

B

A
.

{\displaystyle [A,B] := \mathcal{L}\_A B = - \mathcal{L}\_B A.}

The Lie derivatives are represented by vector fields, as infinitesimal generators of flows (active diffeomorphisms) on *M*. Looking at it the other way around, the group of diffeomorphisms of *M* has the associated Lie algebra structure, of Lie derivatives, in a way directly analogous to the Lie group theory.

### Exterior calculus

Further information: differential form
The exterior calculus allows for a generalization of the gradient, divergence and curl operators.

The bundle of differential forms, at each point, consists of all totally antisymmetric multilinear maps on the tangent space at that point. It is naturally divided into *n*-forms for each *n* at most equal to the dimension of the manifold; an *n*-form is an *n*-variable form, also called a form of degree *n*. The 1-forms are the cotangent vectors, while the 0-forms are just scalar functions. In general, an *n*-form is a tensor with cotangent rank *n* and tangent rank 0. But not every such tensor is a form, as a form must be antisymmetric.

#### Exterior derivative

There is a map from scalars to covectors called the exterior derivative

d

:

C

(
M
)
→

T

∗

(
M
)
:
f
↦

d

f

{\displaystyle \mathrm{d} \colon \mathcal{C}(M) \to \mathrm{T}^\*(M) : f \mapsto \mathrm{d}f}

such that

d

f
:

T

(
M
)
→

C

(
M
)
:
V
↦
V
(
f
)
.

{\displaystyle \mathrm{d}f \colon \mathrm{T}(M) \to \mathcal{C}(M) : V \mapsto V(f).}

This map is the one that relates covectors to infinitesimal displacements, mentioned above; some covectors are the exterior derivatives of scalar functions. It can be generalized into a map from the *n*-forms onto the (*n*+1)-forms. Applying this derivative twice will produce a zero form. Forms with zero derivative are called closed forms, while forms that are themselves exterior derivatives are known as exact forms.

The space of differential forms at a point is the archetypal example of an exterior algebra; thus it possesses a wedge product, mapping a *k*-form and *l*-form to a (*k* + *l*)-form. The exterior derivative extends to this algebra, and satisfies a version of the product rule:

d

(
ω
∧
η
# )

d

ω
∧
η
(
−
1

)

d
e
g

ω

(
ω
∧

d

η
)
.

{\displaystyle \mathrm{d}(\omega \wedge \eta) = \mathrm{d} \omega \wedge \eta+(-1)^{{\rm deg\,}\omega}(\omega \wedge \mathrm{d} \eta).}

From the differential forms and the exterior derivative, one can define the de Rham cohomology of the manifold. The rank *n* cohomology group is the quotient group of the closed forms by the exact forms.

## Topology of differentiable manifolds

### Relationship with topological manifolds

Suppose that

M

{\displaystyle M}

 is a topological

n

{\displaystyle n}

-manifold.

If given any smooth atlas

{
(

U
α

,

φ
α

)

}

α
∈
A

{\displaystyle \{(U\_\alpha,\varphi\_\alpha)\}\_{\alpha\in A}}

, it is easy to find a smooth atlas which defines a different smooth manifold structure on

M
;

{\displaystyle M;}

 consider a homemorphism

Φ
:
M
→
M

{\displaystyle \Phi:M\to M}

 which is not smooth relative to the given atlas; for instance, one can modify the identity map localized non-smooth bump. Then consider the new atlas

{
(

Φ

−
1

(

U
α

)
,

φ
α

∘
Φ
)

}

α
∈
A

,

{\displaystyle \{(\Phi^{-1}(U\_\alpha),\varphi\_\alpha\circ\Phi)\}\_{\alpha\in A},}

 which is easily verified as a smooth atlas. However, the charts in the new atlas are not smoothly compatible with the charts in the old atlas, since this would require that

φ
α

∘
Φ
∘

φ
β

−
1

{\displaystyle \varphi\_\alpha\circ\Phi\circ\varphi\_\beta^{-1}}

 and

φ
α

∘

Φ

−
1

∘

φ
β

−
1

{\displaystyle \varphi\_\alpha\circ\Phi^{-1}\circ\varphi\_\beta^{-1}}

 are smooth for any

α

{\displaystyle \alpha}

 and

β
,

{\displaystyle \beta,}

 with these conditions being exactly the definition that both

Φ

{\displaystyle \Phi}

 and

Φ

−
1

{\displaystyle \Phi^{-1}}

 are smooth, in contradiction to how

Φ

{\displaystyle \Phi}

 was selected.

With this observation as motivation, one can define an equivalence relation on the space of smooth atlases on

M

{\displaystyle M}

 by declaring that smooth atlases

{
(

U
α

,

φ
α

)

}

α
∈
A

{\displaystyle \{(U\_\alpha,\varphi\_\alpha)\}\_{\alpha\in A}}

 and

{
(

V
β

,

ψ
β

)

}

β
∈
B

{\displaystyle \{(V\_\beta,\psi\_\beta)\}\_{\beta\in B}}

 are equivalent if there is a homeomorphism

Φ
:
M
→
M

{\displaystyle \Phi:M\to M}

 such that

{
(

Φ

−
1

(

U
α

)
,

φ
α

∘
Φ
)

}

α
∈
A

{\displaystyle \{(\Phi^{-1}(U\_\alpha),\varphi\_\alpha\circ\Phi)\}\_{\alpha\in A}}

 is smoothly compatible with

{
(

V
β

,

ψ
β

)

}

β
∈
B

,

{\displaystyle \{(V\_\beta,\psi\_\beta)\}\_{\beta\in B},}

 and such that

{
(
Φ
(

V
β

)
,

ψ
β

∘

Φ

−
1

)

}

β
∈
B

{\displaystyle \{(\Phi(V\_\beta),\psi\_\beta\circ\Phi^{-1})\}\_{\beta\in B}}

 is smoothly compatible with

{
(

U
α

,

φ
α

)

}

α
∈
A

.

{\displaystyle \{(U\_\alpha,\varphi\_\alpha)\}\_{\alpha\in A}.}

More briefly, one could say that two smooth atlases are equivalent if there exists a diffeomorphism

M
→
M
,

{\displaystyle M\to M,}

 in which one smooth atlas is taken for the domain and the other smooth atlas is taken for the range.

Note that this equivalence relation is a refinement of the equivalence relation which defines a smooth manifold structure, as any two smoothly compatible atlases are also compatible in the present sense; one can take

Φ

{\displaystyle \Phi}

 to be the identity map.

If the dimension of

M

{\displaystyle M}

 is 1, 2, or 3, then there exists a smooth structure on

M

{\displaystyle M}

, and all distinct smooth structures are equivalent in the above sense. The situation is more complicated in higher dimensions, although it isn't fully understood.

* Some topological manifolds admit no smooth structures, as was originally shown with a ten-dimensional example by Kervaire (1960). A major application of partial differential equations in differential geometry due to Simon Donaldson, in combination with results of Michael Freedman, shows that many simply-connected compact topological 4-manifolds do not admit smooth structures. A well-known particular example is the E8 manifold.
* Some topological manifolds admit many smooth structures which are not equivalent in the sense given above. This was originally discovered by John Milnor in the form of the exotic 7-spheres.

### Classification

Every one-dimensional connected smooth manifold is diffeomorphic to either

R

{\displaystyle \R}

 or

S
1

,

{\displaystyle S^1,}

 each with their standard smooth structures.

For a classification of smooth 2-manifolds, see surface. A particular result is that every two-dimensional connected compact smooth manifold is diffeomorphic to one of the following:

S
2

,

{\displaystyle S^2,}

 or

(

S
1

×

S
1

)
♯
⋯
♯
(

S
1

×

S
1

)
,

{\displaystyle (S^1\times S^1)\sharp\cdots\sharp(S^1\times S^1),}

 or

R
P

2

♯
⋯
♯

R
P

2

.

{\displaystyle \mathbb{RP}^2\sharp\cdots\sharp\mathbb{RP}^2.}

 The situation is more nontrivial if one considers complex-differentiable structure instead of smooth structure.

The situation in three dimensions is quite a bit more complicated, and known results are more indirect. A remarkable result, proved in 2002 by methods of partial differential equations, is the geometrization conjecture, stating loosely that any compact smooth 3-manifold can be split up into different parts, each of which admits Riemannian metrics which possess many symmetries. There are also various "recognition results" for geometrizable 3-manifolds, such as Mostow rigidity and Sela's algorithm for the isomorphism problem for hyperbolic groups.

The classification of *n*-manifolds for *n* greater than three is known to be impossible, even up to homotopy equivalence. Given any finitely presented group, one can construct a closed 4-manifold having that group as fundamental group. Since there is no algorithm to decide the isomorphism problem for finitely presented groups, there is no algorithm to decide whether two 4-manifolds have the same fundamental group. Since the previously described construction results in a class of 4-manifolds that are homeomorphic if and only if their groups are isomorphic, the homeomorphism problem for 4-manifolds is undecidable. In addition, since even recognizing the trivial group is undecidable, it is not even possible in general to decide whether a manifold has trivial fundamental group, i.e. is simply connected.

Simply connected 4-manifolds have been classified up to homeomorphism by Freedman using the intersection form and Kirby–Siebenmann invariant. Smooth 4-manifold theory is known to be much more complicated, as the exotic smooth structures on **R**4 demonstrate.

However, the situation becomes more tractable for simply connected smooth manifolds of dimension ≥ 5, where the h-cobordism theorem can be used to reduce the classification to a classification up to homotopy equivalence, and surgery theory can be applied. This has been carried out to provide an explicit classification of simply connected 5-manifolds by Dennis Barden.

## Structures on smooth manifolds

### (Pseudo-)Riemannian manifolds

A Riemannian manifold consists of a smooth manifold together with a positive-definite inner product on each of the individual tangent spaces. This collection of inner products is called the Riemannian metric, and is naturally a symmetric 2-tensor field. This "metric" identifies a natural vector space isomorphism

T
p

M
→

T
p
∗

M

{\displaystyle T\_pM\to T\_p^\ast M}

 for each

p
∈
M
.

{\displaystyle p\in M.}

 On a Riemannian manifold one can define notions of length, volume, and angle. Any smooth manifold can be given many different Riemannian metrics.

A pseudo-Riemannian manifold is a generalization of the notion of Riemannian manifold where the inner products are allowed to have an indefinite signature, as opposed to being positive-definite; they are still required to be non-degenerate. Every smooth pseudo-Riemannian and Riemmannian manifold defines a number of associated tensor fields, such as the Riemann curvature tensor. Pseudo-Riemannian manifolds of signature (3, 1) are fundamental in general relativity. Not every smooth manifold can be given a (non-Riemannian) pseudo-Riemannian structure; there are topological restrictions on doing so.

A Finsler manifold is a different generalization of a Riemannian manifold, in which the inner product is replaced with a vector norm; as such, this allows the definition of length, but not angle.

### Symplectic manifolds

Further information: symplectic manifold
A symplectic manifold is a manifold equipped with a closed, nondegenerate 2-form. This condition forces symplectic manifolds to be even-dimensional, due to the fact that skew-symmetric

(
2
n
1
)
×
(
2
n
1
)

{\displaystyle (2n+1)\times(2n+1)}

 matrices all have zero determinant. There are two basic examples:

* Cotangent bundles, which arise as phase spaces in Hamiltonian mechanics, are a motivating example, since they have admit a natural symplectic form.
* All oriented two-dimensional Riemannian manifolds

(
M
,
g
)

{\displaystyle (M,g)}

 are, in a natural way, symplectic, by defining the form

ω
(
u
,
v
# )
g
(
u
,
J
(
v
)
)

{\displaystyle \omega(u,v)=g(u,J(v))}

 where, for any

v
∈

T
p

M
,

{\displaystyle v\in T\_pM,}

J
(
v
)

{\displaystyle J(v)}

 denotes the vector such that

v
,
J
(
v
)

{\displaystyle v,J(v)}

 is an oriented

g
p

{\displaystyle g\_p}

-orthonormal basis of

T
p

M
.

{\displaystyle T\_pM.}

### Lie groups

Further information: Lie group
A Lie group consists of a *C*∞ manifold

G

{\displaystyle G}

 together with a group structure on

G

{\displaystyle G}

 such that the product and inversion maps

m
:
G
×
G
→
G

{\displaystyle m:G\times G\to G}

 and

i
:
G
→
G

{\displaystyle i:G\to G}

 are smooth as maps of manifolds. These objects often arise naturally in describing (continuous) symmetries, and they form an important source of examples of smooth manifolds.

Many otherwise familiar examples of smooth manifolds, however, cannot be given a Lie group structure, since given a Lie group

G

{\displaystyle G}

 and any

g
∈
G

{\displaystyle g \in G}

, one could consider the map

m
(
g
,
⋅
)
:
G
→
G

{\displaystyle m(g,\cdot):G\to G}

 which sends the identity element

e

{\displaystyle e}

 to

g

{\displaystyle g}

 and hence, by considering the differential

T
e

G
→

T
g

G
,

{\displaystyle T\_eG\to T\_gG,}

 gives a natural identification between any two tangent spaces of a Lie group. In particular, by considering an arbitrary nonzero vector in

T
e

G
,

{\displaystyle T\_eG,}

 one can use these identifications to give a smooth non-vanishing vector field on

G
.

{\displaystyle G.}

 This shows, for instance, that no even-dimensional sphere can support a Lie group structure. The same argument shows, more generally, that every Lie group must be parallelizable.

## Generalizations

The category of smooth manifolds with smooth maps lacks certain desirable properties, and people have tried to generalize smooth manifolds in order to rectify this. Diffeological spaces use a different notion of chart known as a "plot". Frölicher spaces and orbifolds are other attempts.

A rectifiable set generalizes the idea of a piece-wise smooth or rectifiable curve to higher dimensions; however, rectifiable sets are not in general manifolds.

Banach manifolds and Fréchet manifolds, in particular
manifolds of mappings
are infinite dimensional differentiable manifolds.

### Non-commutative geometry

For a *Ck* manifold *M*, the set of real-valued *Ck* functions on the manifold forms an algebra under pointwise addition and multiplication, called the *algebra of scalar fields* or simply the *algebra of scalars*. This algebra has the constant function 1 as the multiplicative identity, and is a differentiable analog of the ring of regular functions in algebraic geometry.

It is possible to reconstruct a manifold from its algebra of scalars, first as a set, but also as a topological space – this is an application of the Banach–Stone theorem, and is more formally known as the spectrum of a C\*-algebra. First, there is a one-to-one correspondence between the points of *M* and the algebra homomorphisms *φ*: *Ck*(*M*) → **R**, as such a homomorphism *φ* corresponds to a codimension one ideal in *Ck*(*M*) (namely the kernel of *φ*), which is necessarily a maximal ideal. On the converse, every maximal ideal in this algebra is an ideal of functions vanishing at a single point, which demonstrates that MSpec (the Max Spec) of *Ck*(*M*) recovers *M* as a point set, though in fact it recovers *M* as a topological space.

One can define various geometric structures algebraically in terms of the algebra of scalars, and these definitions often generalize to algebraic geometry (interpreting rings geometrically) and operator theory (interpreting Banach spaces geometrically). For example, the tangent bundle to *M* can be defined as the derivations of the algebra of smooth functions on *M*.

This "algebraization" of a manifold (replacing a geometric object with an algebra) leads to the notion of a C\*-algebra – a commutative C\*-algebra being precisely the ring of scalars of a manifold, by Banach–Stone, and allows one to consider *non*commutative C\*-algebras as non-commutative generalizations of manifolds. This is the basis of the field of noncommutative geometry.