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Definition 1 (Topological Manifold)

A topological space M is called an n-dimensional topological manifold (or n-manifold) if,

  1. M is Hausdorff.
  2. M is second-countable.
  3. Every point x \in M has an open neighbourhood U \subset M that is homeomorphic to an open subset of \mathbb{R}^n.

Note: As a convention, the ball B^0 is a single point. Any space with the discrete topology is a 0-dimensional manifold.

Note also that all topological manifolds are clearly locally connected.

To emphasize that a given manifold M is n-dimensional, we will use the shorthand M^n. This is not to be confused with an n-ary cartesian product M\times...\times M. However, we will prove later that such a construction does exist as well.

The alert reader may wonder why we require the manifold to be Hausdorff and second-countable. The reason for this is to exclude some pathological examples. Two such examples are the long line, which is not second-countable, and the line with two origins, which is not Hausdorff.

Theorem 2

A topological manifold is connected if and only if it is pathwise connected.

Proof: Since all topological manifolds are clearly locally connected, the theorem immediately follows.

Definition 3

Let M be a topological n-manifold. Let p\in M and p\in U\subseteq M be an open neighborhood of p. Now, let \phi\,:\, U \rightarrow U^\prime, where U^\prime \subseteq \mathbb{R}^n, be a homeomorphism. Then the pair (U,\phi) is called a chart at p.

Definition 4

Let M be a topological n-manifold, and let A=\{(U_i,\phi_i)\}_{i\in I} be charts on M such that \bigcup_{i\in I} U_i = M. That is, \{U_i\}_{i\in I} is an open covering of M. Then A is called an atlas on M.

Definition 5

Let M be an n-manifold and let (U,\phi) and (V,\psi) be charts are a point p (so U\cap V \neq \emptyset). Define the transition function (or chart transformation) between the two charts as the homeomorphism \psi\circ \phi^{-1} : \phi(U\cap V) \rightarrow \psi(U\cap V).

Given a pair (M,A), where M is an n-manifold and A is an atlas on M, properties that M may satifsy are often expressed as properties of the transition functions between charts in A. This is how we will define our notion of a differentiable manifold.

Definition 6

An atlas for a manifold is smooth (or \mathcal{C}^\infty) if all the transition functions are smooth (All higher order partial derivatives exist and are continuous).

Definition 7

A diffeomorphism is a smooth homeomorphism f such that f^{-1} is also smooth.

Note that in a smooth atlas, all transition functions are diffeomorphisms.

Definition 8

Let M be a manifold and A be a smooth atlas on M. Then, define A_{\mathrm{max}} as the set of all charts (V,\psi) on M such that for all (U,\phi)\in A, \psi\circ \phi^{-1}|_{\phi(U\cap V)} and \psi\circ \phi^{-1}|_{\phi(U\cap V)} are smooth.

A chart (V,\psi) with the property described above is said to be compatible with A.

Lemma 9

A_{max} is a smooth atlas on M.

Proof: We have to show that the transition functions between any pair of charts in A_{\mathrm{max}} are smooth. This is obvious if one of then is inA, so let (V_1,\psi_1) and (V_2,\psi_2) be charts in A_{\mathrm{max}} that are not in A, such that V_1\cap V_2\neq \emptyset. Let (U,\phi)\in A be a chart such that U\cap V_1 \cap V_2 = W\neq \emptyset. Then \phi\circ \psi_1^{-1}|_W and \psi_2\circ \phi^{-1}|_W are both smooth, since, both (V_1,\psi_1) and (V_2,\psi_2) are compatible with A. Then, \psi_2\circ \psi_1^{-1}|_W=\psi_2\circ (\phi^{-1}\circ\phi )\circ \psi_1^{-1}|_W=(\psi_2\circ \phi^{-1}|_W) \circ (\phi\circ \psi_1^{-1}|_W) is smooth since it is a composition of smooth maps. An identical argument for \psi_1\circ \psi_2^{-1} completes the proof.

It should be obvious that if A^\prime is a smooth atlas containing a smooth atlas A, then A_{\mathrm{max}}^\prime=A_{\mathrm{max}}.

Smooth maps[edit]

Definition 10

Let M^m N^n be smooth manifolds, p\in M, and let f\,:\, M\rightarrow N be a function. Then, if for any charts (U,\phi) on M and (V,\psi) on N such that p\in U and f(p)\in V, the function \hat{f}(\phi(p))=(\psi\circ f\circ \phi^{-1})(\psi(p)) from \phi(U\cap f^{-1}(V))\subseteq \mathbb{R}^m to \psi(f(U)\cap V)\subseteq \mathbb{R}^n is a smooth function on euclidean spaces, then f\,:\, M\rightarrow N is said to be smooth at p. f is called a smooth function if it is smooth for all p\in M.

Lemma 11

f\,:\, M\rightarrow N is smooth at p\in M if and only if there exists charts (U,\phi) on M and (V,\psi) on N with p\in U\subseteq f^{-1}(V) such that (\psi\circ f\circ\phi^{-1}):\phi(U\cap f^{-1}(V)) \rightarrow \psi(f(U)\cap V) is smooth.

Proof: f is continuous since \psi\circ f\circ\phi^{-1} is smooth and thus continuous and \phi and \psi are homeomorphisms. Let (U^\prime,\phi^\prime) and (V^\prime,\psi^\prime) be two other charts at p and f(p). Then (\psi^\prime\circ f\circ{\phi^\prime}^{-1})=(\psi^\prime\circ\psi^{-1})\circ (\psi\circ f \circ \phi^{-1})\circ (\phi\circ {\phi^\prime}^{-1}) which is a composition of smooth functions since the atlases on M and N are smooth, and is therefore smooth.

Remark 12

By Lemma 11, we do not have to check all charts to see if a function is smooth. A relief, since maximal atlases tend to be uncountably big.

Definition 12

If f\,:\,M\rightarrow N is a smooth bijective function such that its inverse is smooth too, it is called a diffeomorphism. Two manifolds are called diffeomorphic if there exists a diffeomeorphism between them.

Lemma 13

Let f\,:\,M\rightarrow N and g\,:\,N\rightarrow P be smooth. Then g\circ f \,:\, M\rightarrow P is smooth as well.

Proof: Let (U,\phi), (V,\psi) and (W,\xi) be charts on M,N,P at p,f(p),g\circ f(p) respectively. Then \xi \circ g\circ f\circ \phi^{-1}(\phi(p))=(\xi\circ g\circ\psi^{-1})\circ (\psi\circ f\circ\phi^{-1})(\phi(p)) which is a composition of smooth maps of euclidean space and is hence smooth.

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