nLab differentiable manifold

Contents

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

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

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

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 $X$, “smooth curves” in $X$. 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 $x \in X$ constitute the tangent space $T_x X$, and the collection of all these tangent spaces yields another differentiable manifold, called the tangent bundle $T X$. This happens to be a vector bundle which is associated to a principal bundle, called the frame bundle $Fr(X)$.

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

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

Remark

(varying terminology)

Often a topological manifold (def. ) 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 $n$-dimensional topological manifold $X$ (def. ), then 1. an open subset $U \subset X$ and a homeomorphism $\phi \colon \mathbb{R}^n \overset{\phantom{A}\simeq\phantom{A}}{\to} U$ is also called a local coordinate chart of $X$.

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

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

$U_{i j} \coloneqq U_i \cap U_j$

then the induced homeomorphism

$\mathbb{R}^n \supset \phantom{AA} \phi_i^{-1}(U_{i j}) \overset{\phantom{A}\phi_i\phantom{A}}{\longrightarrow} U_{i j} \overset{\phantom{A}\phi_j^{-1}\phantom{A}}{\longrightarrow} \phi_j^{-1}(U_{i j}) \phantom{AA} \subset \mathbb{R}^n$

is called the gluing function from chart $i$ to chart $j$.

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

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

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

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

We say that $f$ is differentiable on a subset $I$ of $U$ if $f$ is differentiable at every $x\in I$, and differentiable if $f$ is differentiable on all of $U$. We say that $f$ is continuously differentiable if it is differentiable and $d f$ is a continuous function.

The map $d f_x$ is called the derivative or differential of $f$ at $x$.

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

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

$f_i \;\colon\; U \overset{f}{\longrightarrow} \mathbb{R}^{n_2} \overset{pr_u}{\longrightarrow} \mathbb{R}$

is differentiable in the previous sense

In this case, the derivatives $d f_i \colon \mathbb{R}^n \to \mathbb{R}$ of the $f_i$ assemble into a linear map of the form

$d f_x \;\colon\; \mathbb{R}^{n_1} \to \mathbb{R}^{n_2} \,.$

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

$d f \;\colon\; U \longrightarrow Hom_{\mathbb{R}}(\mathbb{R}^{n_1} , \mathbb{R}^{n_2}) \simeq \mathbb{R}^{n_1 \cdot n_2}$

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

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

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

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 $n_1, n_2, n_3 \in \mathbb{N}$ and let

$\mathbb{R}^{n_1} \overset{f}{\longrightarrow} \mathbb{R}^{n_2} \overset{g}{\longrightarrow} \mathbb{R}^{n_3}$

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

$d(g \circ f) = (d g) \circ (d f)$

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

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

Differentiable manifolds

Definition

(differentiable manifold and smooth manifold)

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

1. a topological manifold $X$ (def. );

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

A $p$-fold differentiable function between $p$-fold differentiable manifolds

$\left(X,\, \{\mathbb{R}^{n} \overset{\phi_i}{\to} U_i \subset X\}_{i \in I} \right) \overset{\phantom{AA}f\phantom{AA}}{\longrightarrow} \left(Y,\, \{\mathbb{R}^{n'} \overset{\psi_j}{\to} V_j \subset Y\}_{j \in J} \right)$

is

such that

• for all $i \in I$ and $j \in J$ then

$\mathbb{R}^n \supset \phantom{AA} (f\circ \phi_i)^{-1}(V_j) \overset{\phi_i}{\longrightarrow} f^{-1}(V_j) \overset{f}{\longrightarrow} V_j \overset{\psi_j^{-1}}{\longrightarrow} \mathbb{R}^{n'}$

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

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

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

Remark

(category Diff of differentiable manifolds)

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

The analog of the concept of homeomorphism (def. ) is now this:

Definition

(diffeomorphism)

Given smooth manifolds $X$ and $Y$ (def. ), then a smooth function

$f \;\colon\; X \longrightarrow Y$

is called a diffeomorphism, if there is an inverse function

$X \longleftarrow Y \;\colon\; g$

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

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 $X$ be a topological manifold (def. ) and let

$\left( \mathbb{R}^n \underoverset{\simeq}{\phi_i}{\longrightarrow} U_i \subset X \right)_{i \in I} \phantom{AAA} \text{and} \phantom{AAA} \left( \mathbb{R}^{n} \underoverset{\simeq}{\psi_j}{\longrightarrow} V_j \subset X \right)_{j \in J}$

be two atlases (def. ), both making $X$ into a smooth manifold (def. ).

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

$f \;\colon\; \left( X \;,\; \left( \mathbb{R}^n \underoverset{\simeq}{\phi_i}{\longrightarrow} U_i \subset X \right)_{i \in I} \right) \overset{\simeq}{\longrightarrow} \left( X\;,\; \left( \mathbb{R}^{n} \underoverset{\simeq}{\psi_j}{\longrightarrow} V_j \subset X \right)_{j \in J} \right)$

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

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

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

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

$\mathbb{R}^n \supset \, \phi_i^{-1}(U_i \cap U_j) \overset{\phi_i}{\longrightarrow} U_i \cap V_j \overset{\psi_j^{-1}}{\longrightarrow} \psi_j^{-1}(U_i \cap V_j) \; \subset \mathbb{R}^n$

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 $X$. An equivalence class with respect to this equivalence relation is called a smooth structure on $X$.

As Étale Spaces

Differentiable manifolds can be viewed as akin to étale spaces, very much in alignment with the language of algebraic geometry. For smooth manifolds and complex manifolds the requisite atlas conditions can be stated in terms of a particular structure sheaf. More precisely, this definition yields an equivalence of categories between the category of smooth manifolds as in and the category of locally ringed spaces modeled on domains of $\mathbf{R}^n$ with a sheaf of germs of smooth functions. The same goes for topological and complex manifolds, where the local models are domains with sheafs of germs of continuous and holomorphic functions respectively. Here we illustrate the differentiable case. For any $A \subset {\mathbf R}^n$, we write $\mathcal{D}_A$ for the sheaf of germs of functions $f \in C^{\infty}(A, \mathbf{R} )$. That is, $\mathcal{D}_A$ is the functor $\mathcal{D}_A : \text{Open} (A) \rightarrow {\mathbf Ring}^{\text{op}}$ s.t. $\mathcal{D}_A : B \mapsto C^{\infty}(B, {\mathbf R})$ and $\mathcal{D}_A : B \subset B' \mapsto C^{\infty}(B, {\mathbf R}) \hookrightarrow C^{\infty}(B', {\mathbf R})$. This turns $(A, \mathcal{D}_A)$ into a ringed space, and any ringed space modeled on one of this kind can be regarded as a differentiable (by which is always meant “real”) manifold. More precisely, given a ringed space $(X, D_X)$, there must an open set $A \subset {\mathbf R}^n$ and local isomorphism $(f, f_*)$ where $f: A \rightarrow X$ is a continuous function with sheaf pushfoward $f_* : (\mathcal{O}_A \rightarrow \mathcal{O}_X )$. The following property, which holds at the continuous level, is essential.

Lemma

Let $F : X \rightarrow Y$ be a continuous surjection. Let $P$ be a sheaf of $R$-valued functions on $X$ (here we let $R$ be any ring). Consider an open $U \subset Y$ and define $Q(U) \subset R^U$ to be $\{ f : U \rightarrow R | F^{*} f = f \circ F|_{f^{-1}(U)} \in P(F^{-1}(U)) \}$ i.e. those $f$ s.t. the pullback of each $f$ by $F$ belongs to $P(F^{-1}(U))$. Then $Q$ is a sheaf on $Y$.

We can phrase the example of $\mathbf{R}P^n$ in this language.

Example

(real projective space as a smooth manifold in this étale map sense)

We need only show that the functions $f : V \subset {\mathbf R}P^n \rightarrow {\mathbf R}^n$ which are smooth when composed with the canonical projection $\pi : {\mathbf R}^{n+1}/\{ 0 \} \rightarrow {\mathbf R }P^n$ form a sheaf on ${\mathbf R}^n$. Denote these $\mathcal{O}_{{\mathbf R}P^n}$. Certainly they form a presheaf, and checking the pullback as in the lemma confirms that they are a sheaf. So far this only confirms that ${\mathbf R}P^n$ is a real topological manifold. To establish its differentiable structure we take for each chart $(U, \varphi)$ of ${\mathbf R}P^n$ the ringed space $(\varphi(U), \varphi_* \mathcal{O}_{{\mathbf R}P^n}|_U )$ obtained from restriction and then pushforward, a local isomorphism of locally ringed spaces by construction. Hence $({\mathbf R}P^n, \mathcal{O}_{{\mathbf R}P^n})$ is a smooth manifold.

Examples

Example

(Cartesian space as a smooth manifold)

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

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

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

Example

(n-spheres as smooth manifolds)

For all $n \in \mathbb{N}$, the n-sphere $S^n$ 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

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

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

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

1. $S^n \subset \mathbb{R}^{n+1}$ 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 $n$-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 $n \in \mathbb{N}$

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

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

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 $X$ any $Y$ be two differentiable manifolds with atlases $\{\mathbb{R}^n\underoverset{\simeq}{\phi_i}{\to} U_i \subset X\}$ and $\{\mathbb{R}^{n'}\underoverset{\simeq}{\psi_j}{\to} V_j \subset Y\}$. Then their product topological space $X \times Y$ becomes an differentiable manifold with respect to the atlas

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

(open subsets of differentiable manifolds are differentiable manifolds)

Let $X$ be a $k$-fold differentiable manifold and let $S \subset X$ be an open subset of the underlying topological space $(X,\tau)$.

Then $S$ carries the structure of a $k$-fold differentiable manifold such that the inclusion map $S \hookrightarrow X$ is an open embedding of differentiable manifolds.

Proof

Since the underlying topological space of $X$ 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 $X$ has a single connected component. By this prop this implies that $X$ 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 $S$ is still locally Euclidean: since $X$ is locally Euclidean every point $x \in S \subset X$ has a Euclidean neighbourhood in $X$ and since $S$ is open there exists an open ball in that (itself homeomorphic to Euclidean space) which is a Euclidean neighbourhood of $x$ contained in $S$.

For the differentiable structure we pick these Euclidean neighbourhoods from the given atlas. Then the gluing functions for the Euclidean charts on $S$ are $k$-fold differentiable follows since these are restrictions of the gluing functions for the atlas of $X$.

Example

(general linear group)

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

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

Textbook accounts include

Lecture notes include