nLab
differentiable manifold

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Manifolds and cobordisms

Contents

Idea

A differentiable manifold is a topological space which is locally homeomorphic to a Euclidean space (a topological manifold) and such that the gluing functions which relate these Euclidean local charts to each other are differentiable functions, for a fixed degree of differentiability. If one considers arbitrary differentiablity then one speaks of smooth manifolds and if one demands analytic gluing functions then one speaks of analytic manifolds. For a general discussion see at manifold.

Accordingly, a differentiable manifold is a space to which the tools of (infinitesimal) analysis may be applied locally. Notably we may ask whether a continuous function between differentiable manifolds is differentiable by computing its derivatives pointwise in any of the Euclidean coordinate charts.

In particular one may consider smooth functions from the real line into any smooth manifold XX, “smooth curves” in XX. The equivalence classes of these that have the same first derivative at a given point capture the idea of “infinitsimal smooth paths” through that point in the manifold, called its tangent vectors. All the tangent vectors at one point xXx \in X constitute the tangent space T xXT_x X, and the collection of all these tangent spaces yields another differentiable manifold, called the tangent bundle TXT X. This happens to be a vector bundle which is associated to a principal bundle, called the frame bundle Fr(X)Fr(X).

By equipping the tangent bundle or frame bundle of a differentiable manifold with extra properties or extra structure one encodes geometry on the manifold. For example equipping them with orthogonal structure encodes Riemannian geometry on manifolds. More generally one may consider any G-structure on the frame bundle and thereby equip the manifold with the corresponding Cartan geometry, for instance complex geometry, conformal geometry etc.

This way differential and smooth manifolds are the basis for differential geometry. They are the analogs in differential geometry of what schemes are in algebraic geometry. In fact both of these concepts are unified within synthetic differential geometry.

If one relaxes the condition on differentiable manifold from it being locally isomorphic to a Euclidean space to it just admitting local smooth maps from a Euclidean space, then one obtains the more general concept of diffeological spaces or even smooth sets, see at generalized smooth space for more on this. If one generalizes here differentiable functions to simplicial differentiable functions one obtains concepts of derived smooth manifold.

The generalization of differentiable manifolds to higher differential geometry are orbifolds and more generally differentiable stacks. If one combines this with the generalization to smooth sets then one obtains the concept of smooth stacks and eventually smooth infinity-stacks.

Definition

For convenience, we first recall the basic definition of

and of

and then turn to the actual definition of

Topological manifolds

For convenience, we first recall here some background on topological manifolds:

Definition

(topological manifold)

A topological manifold is a topological space which is

  1. locally Euclidean,

  2. paracompact Hausdorff.

If the local Euclidean neighbourhoods nUX\mathbb{R}^n \overset{\simeq}{\to} U \subset X are all of dimension nn for a fixed nn \in \mathbb{N}, then the topological manifold is said to be a nn-dimensional manifold or nn-fold. This is usually assumed to be the case.

Remark

(varying terminology)

Often a topological manifold (def. 1) is required to be sigma-compact. But by this prop. this is not an extra condition as long as there is a countable set of connected components. Moreover, manifolds with uncountably many connected components are rarely considered in practice.

Definition

(local chart, atlas and gluing function)

Given an nn-dimensional topological manifold XX (def. 1), then

  1. an open subset UXU \subset X and a homeomorphism ϕ: nAAU\phi \colon \mathbb{R}^n \overset{\phantom{A}\simeq\phantom{A}}{\to} U is also called a local coordinate chart of XX.

  2. an open cover of XX by local charts { nϕ iUX} iI\left\{ \mathbb{R}^n \overset{\phi_i}{\to} U \subset X \right\}_{i \in I} is called an atlas of the topological manifold.

  3. denoting for each i,jIi,j \in I the intersection of the iith chart with the jjth chart in such an atlas by

    U ijU iU j U_{i j} \coloneqq U_i \cap U_j

    then the induced homeomorphism

    nAAϕ i 1(U ij)Aϕ iAU ijAϕ j 1Aϕ j 1(U ij)AA n \mathbb{R}^n \supset \phantom{AA} \phi_i^{-1}(U_{i j}) \overset{\phantom{A}\phi_i\phantom{A}}{\longrightarrow} U_{i j} \overset{\phantom{A}\phi_j^{-1}\phantom{A}}{\longrightarrow} \phi_j^{-1}(U_{i j}) \phantom{AA} \subset \mathbb{R}^n

    is called the gluing function from chart ii to chart jj.

graphics grabbed from Frankel

Differentiable functions between Cartesian spaces

For convenience we recall the definition of differentiable functions between Euclidean spaces.

Definition

(differentiable functions between Euclidean spaces)

Let nn \in \mathbb{N} and let U nU \subset \mathbb{R}^n be an open subset.

Then a function f:Uf \;\colon\; U \longrightarrow \mathbb{R} is called differentiable at xUx\in U if there exists a linear map df x: nd f_x : \mathbb{R}^n \to \mathbb{R} such that the following limit exists as hh approaches zero “from all directions at once”:

lim h0f(x+h)f(x)df x(h)h=0. \lim_{h\to 0} \frac{f(x+h)-f(x) - d f_x(h)}{\Vert h\Vert} = 0.

This means that for all ϵ(0,)\epsilon \in (0,\infty) there exists an open subset VUV\subseteq U containing xx such that whenever x+hVx+h\in V we have f(x+h)f(x)df x(h)h<ϵ\frac{f(x+h)-f(x) - d f_x(h)}{\Vert h\Vert} \lt \epsilon.

We say that ff is differentiable on a subset II of UU if ff is differentiable at every xIx\in I, and differentiable if ff is differentiable on all of UU. We say that ff is continuously differentiable if it is differentiable and dfd f is a continuous function.

The map df xd f_x is called the derivative or differential of ff at xx.

More generally, let n 1,n 2n_1, n_2 \in \mathbb{N} and let U n 1U\subseteq \mathbb{R}^{n_1} be an open subset.

Then a function f:U n 2f \;\colon\; U \longrightarrow \mathbb{R}^{n_2} is differentiable if for all i{1,,n 2}i \in \{1, \cdots, n_2\} the component function

f i:Uf n 2pr u f_i \;\colon\; U \overset{f}{\longrightarrow} \mathbb{R}^{n_2} \overset{pr_u}{\longrightarrow} \mathbb{R}

is differentiable in the previous sense

In this case, the derivatives df i: nd f_i \colon \mathbb{R}^n \to \mathbb{R} of the f if_i assemble into a linear map of the form

df x: n 1 n 2. d f_x \;\colon\; \mathbb{R}^{n_1} \to \mathbb{R}^{n_2} \,.

If the derivative exists at each xUx \in U, then it defines itself a function

df:UHom ( n 1, n 2) n 1n 2 d f \;\colon\; U \longrightarrow Hom_{\mathbb{R}}(\mathbb{R}^{n_1} , \mathbb{R}^{n_2}) \simeq \mathbb{R}^{n_1 \cdot n_2}

to the space of linear maps from n 1\mathbb{R}^{n_1} to n 2\mathbb{R}^{n_2}, which is canonically itself a Euclidean space. We say that ff is twice continuously differentiable if dfd f is continuously differentiable.

Generally then, for kk \in \mathbb{N} the function ff is called kk-fold continuously differentiable or of class C kC^k if the kk-fold differential d kfd^k f exists and is a continuous function.

Finally, if ff is kk-fold continuously differentiable for all kk \in \mathbb{N} then it is called a smooth function or of class C C^\infty.

Of the various properties satisfied by differentiation, the following plays a special role in the theory of differentiable manifolds (notably in the discussion of their tangent bundles):

Proposition

(chain rule for differentiable functions between Euclidean spaces)

Let n 1,n 2,n 3n_1, n_2, n_3 \in \mathbb{N} and let

n 1f n 2g n 3 \mathbb{R}^{n_1} \overset{f}{\longrightarrow} \mathbb{R}^{n_2} \overset{g}{\longrightarrow} \mathbb{R}^{n_3}

be two differentiable functions (def. 3). Then the derivative of their composite is the composite of their derivatives:

d(gf)=(dg)(df) d(g \circ f) = (d g) \circ (d f)

hence for all x n 1x \in \mathbb{R}^{n_1} we have

d(gf) x=dg f(x)df x. d(g \circ f)_x = d g_{f(x)} \circ d f_x \,.

Differentiable manifolds

Definition

(differentiable manifold and smooth manifold)

For p{}p \in \mathbb{N} \cup \{\infty\} then a pp-fold differentiable manifold or C pC^p-manifold for short is

  1. a topological manifold XX (def. 1);

  2. an atlas { nϕ iX}\{\mathbb{R}^n \overset{\phi_i}{\to} X\} (def. 2) all whose gluing functions are pp times continuously differentiable.

A pp-fold differentiable function between pp-fold differentiable manifolds

(X,{ nϕ iU iX} iI)AAfAA(Y,{ nψ jV jY} jJ) \left(X,\, \{\mathbb{R}^{n} \overset{\phi_i}{\to} U_i \subset X\}_{i \in I} \right) \overset{\phantom{AA}f\phantom{AA}}{\longrightarrow} \left(Y,\, \{\mathbb{R}^{n'} \overset{\psi_j}{\to} V_j \subset Y\}_{j \in J} \right)

is

such that

  • for all iIi \in I and jJj \in J then

    nAA(fϕ i) 1(V j)ϕ if 1(V j)fV jψ j 1 n \mathbb{R}^n \supset \phantom{AA} (f\circ \phi_i)^{-1}(V_j) \overset{\phi_i}{\longrightarrow} f^{-1}(V_j) \overset{f}{\longrightarrow} V_j \overset{\psi_j^{-1}}{\longrightarrow} \mathbb{R}^{n'}

    is a pp-fold differentiable function between open subsets of Euclidean space.

(Notice that this in in general a non-trivial condition even if X=YX = Y and ff is the identity function. In this case the above exhibits a passage to a different, but equivalent, differentiable atlas.)

If a manifold is C pC^p differentiable for all pp, then it is called a smooth manifold. Accordingly a continuous function between differentiable manifolds which is pp-fold differentiable for all pp is called a smooth function,

Remark

(category Diff of differentiable manifolds)

In analogy to remark \ref{TopCategory} there is a category called Diff p{}_p (or similar) whose objects are C pC^p-differentiable manifolds and whose morphisms are C pC^p-differentiable functions, for given p{}p \in \mathbb{N} \cup \{\infty\}.

The analog of the concept of homeomorphism (def. \ref{Homeomorphism}) is now this:

Definition

(diffeomorphism)

Given smooth manifolds XX and YY (def. 4), then a smooth function

f:XY f \;\colon\; X \longrightarrow Y

is called a diffeomorphism, if there is an inverse function

XY:g X \longleftarrow Y \;\colon\; g

which is also a smooth function (hence if ff is an isomorphism in the category Diff {}_\infty from remark 2).

Here it is important to note that while being a topological manifold is just a property of a topological space, a differentiable manifold carries extra structure encoded in the atlas:

Definition

(smooth structure)

Let XX be a topological manifold (def. 1) and let

( nϕ iU iX) iIAAAandAAA( nψ jV jX) jJ \left( \mathbb{R}^n \underoverset{\simeq}{\phi_i}{\longrightarrow} U_i \subset X \right)_{i \in I} \phantom{AAA} \text{and} \phantom{AAA} \left( \mathbb{R}^{n} \underoverset{\simeq}{\psi_j}{\longrightarrow} V_j \subset X \right)_{j \in J}

be two atlases (def. 2), both making XX into a smooth manifold (def. 4).

Then there is a diffeomorphism (def. 5) of the form

f:(X,( nϕ iU iX) iI)(X,( nψ jV jX) jJ) f \;\colon\; \left( X \;,\; \left( \mathbb{R}^n \underoverset{\simeq}{\phi_i}{\longrightarrow} U_i \subset X \right)_{i \in I} \right) \overset{\simeq}{\longrightarrow} \left( X\;,\; \left( \mathbb{R}^{n} \underoverset{\simeq}{\psi_j}{\longrightarrow} V_j \subset X \right)_{j \in J} \right)

precisely if the identity function on the underlying set of XX constitutes such a diffeomorphism. (Because if ff is a diffeomorphism, then also f 1f=id Xf^{-1}\circ f = id_X is a diffeomorphism.)

That the identity function is a diffeomorphism between XX equipped with these two atlases means, by definition 4, that

iIjJ(ϕ i 1(V j)ϕ iV jψ j 1 nAAis smooth). \underset{{i \in I} \atop {j \in J}}{\forall} \left( \phi_i^{-1}(V_j) \overset{\phi_i}{\longrightarrow} V_j \overset{\psi_j^{-1}}{\longrightarrow} \mathbb{R}^n \phantom{AA} \text{is smooth} \right) \,.

Notice that the functions on the right may equivalently be written as

nϕ i 1(U iU j)ϕ iU iV jψ j 1ψ j 1(U iV j) n \mathbb{R}^n \supset \, \phi_i^{-1}(U_i \cap U_j) \overset{\phi_i}{\longrightarrow} U_i \cap V_j \overset{\psi_j^{-1}}{\longrightarrow} \psi_j^{-1}(U_i \cap V_j) \; \subset \mathbb{R}^n

showing their analogy to the glueing functions within a single atlas spring.

Hence diffeomorphism induces an equivalence relation on the set of smooth atlases that exist on a given topological manifold XX. An equivalence class with respect to this equivalence relation is called a smooth structure on XX.

Examples

Example

(Cartesian space as a smooth manifold)

For nn \in \mathbb{N} then the Cartesian space n\mathbb{R}^n equipped with the atlas consisting of the single chart nid n\mathbb{R}^n \overset{id}{\to} \mathbb{R}^n is a smooth manifold, in particularly a pp-fold differentiable manifold for every pp \in \mathbb{N} according to def. 4.

Similarly the open disk D nD^n becomes a smooth manifold when equipped with the atlas whose single chart is the homeomorphism nD n\mathbb{R}^n \to D^n.

This defines a smooth structure (def. 6) on n\mathbb{R}^n and D nD^n. Strikingly, precisely for n=4n = 4 there are other smooth structures on 4\mathbb{R}^4, hence called exotic smooth structures.

Example

(n-spheres as smooth manifolds)

For all nn \in \mathbb{N}, the n-sphere S nS^n becomes a smooth manfold, with atlas consisting of the two local charts that are given by the inverse functions of the stereographic projection from the two poles of the sphere onto the equatorial hyperplane

{ nσ i 1S n} i{+,}. \left\{ \mathbb{R}^n \underoverset{\simeq}{\sigma^{-1}_i}{\longrightarrow} S^n \right\}_{i \in \{+,-\}} \,.

By the formulas given in this prop. the induced gluing function n\{0} n\{0}\mathbb{R}^n \backslash \{0\} \to \mathbb{R}^n \backslash \{0\} is a rational function, and hence a smooth function.

Finally the nn-sphere is a paracompact Hausdorff topological space. Ways to see this include:

  1. S n n+1S^n \subset \mathbb{R}^{n+1} is a compact subspace by the Heine-Borel theorem. Compact spaces are evidently also paracompact. Moreover, Euclidean space, like any metric space, is Hausdorff, and subspaces of Hausdorff spaces are Hausdorff;

  2. The nn-sphere has an evident structure of a CW-complex and CW-complexes are paracompact Hausdorff spaces.

Example

(real projective space and complex projective space)

For nn \in \mathbb{N}

  1. the real projective space P n\mathbb{R}P^n has the structure of a smooth manifold of dimension nn;

  2. the complex projective space P n\mathbb{C}P^n has the structure of a smooth manifold of (real) dimension 2n2n.

where in both cases the atlas is given by the standard open cover (this def.).

By this prop..

We now discuss some general mechanisms by which new differentiable manifolds arise from given ones:

Example

(product manifold)

Let XX any YY be two differentiable manifolds with atlases { nϕ iU iX}\{\mathbb{R}^n\underoverset{\simeq}{\phi_i}{\to} U_i \subset X\} and { nψ jV jY}\{\mathbb{R}^{n'}\underoverset{\simeq}{\psi_j}{\to} V_j \subset Y\}. Then their product topological space X×YX \times Y becomes an differentiable manifold with respect to the atlas

{ nnϕ i×ψ jU i×V jX×Y} (i,j)I×J \left\{ \mathbb{R}^{n \cdot n'} \underoverset{\simeq}{\phi_i \times \psi_j}{\to} U_i \times V_j \subset X \times Y \right\}_{(i,j) \in I \times J}
Example

(open subsets of differentiable manifolds are again differentiable manifolds)

Let XX be a kk-fold differentiable manifold and let SXS \subset X be an open subset of the underlying topological space (X,τ)(X,\tau).

Then SS carries the structure of a kk-fold differentiable manifold such that the inclusion map SXS \hookrightarrow X is an open embedding of differentiable manifolds.

Proof

Since the underlying topological space of XX is locally connected (this prop.) it is the disjoint union space of its connected components (this prop.).

Therefore we are reduced to showing the statement for the case that XX has a single connected component. By this prop this implies that XX is second-countable topological space.

Now a subspace of a second-countable Hausdorff space is clearly itself second countable and Hausdorff.

Similarly it is immediate that SS is still locally Euclidean: since XX is locally Euclidean every point xSXx \in S \subset X has a Euclidean neighbourhood in XX and since SS is open there exists an open ball in that (itself homeomorphic to Euclidean space) which is a Euclidean neighbourhood of xx contained in SS.

For the differentiable structure we pick these Euclidean neighbourhoods from the given atlas. Then the gluing functions for the Euclidean charts on SS are kk-fold differentiable follows since these are restrictions of the gluing functions for the atlas of XX.

Example

(general linear group)

For nn \in \mathbb{N}, the general linear group Gl(n,)Gl(n,\mathbb{R}) is a smooth manifold (as an open subspace of Euclidean space GL(n,)Mat n×n() (n 2)GL(n,\mathbb{R}) \subset Mat_{n \times n}(\mathbb{R}) \simeq \mathbb{R}^{(n^2)}, via example 5 and example 1).

The group operations are smooth functions with respect to this smooth manifold structure, and thus GL(n,)GL(n,\mathbb{R}) is a Lie group.

References

Textbook accounts include

Lecture notes include

See also

Revised on June 26, 2017 07:22:12 by Urs Schreiber (88.77.226.246)