1 Manifolds

#math

Intro

Definition

An n-dimensional differentiable manifold $M$ is:

a Hausdorff Topological space.
Equipped with a countable family of pairs $(U_{i}, ϕ_{i})$ with $i \in I$ consisting of a countable family of open sets $U_{i}$ covering $M$ and of countable collection of homeomorphisms $ϕ_{i} : U_{i} \to O_{i} \subset R^{n}$ . Moreover, any two distinct charts $(U_{i}, ϕ_{i})$ and $(U_{j}, ϕ_{j})$ are pairwise compatible, i.e. given $U_{i} \cap U_{j} \neq \emptyset$ , the transition map:

ψ_{i j} = ϕ_{i} \circ ϕ_{j}^{- 1} : ϕ_{j} (U_{i} \cap U_{j}) ⟶ ϕ_{i} (U_{i} \cap U_{j})

is $C^{\infty}$ on its domain definition.
The chart allows us to give the manifold a coordinate representation!
The set of all charts is called an atlas.

Definition

Let $M$ and $N$ be differentiable manifolds of dimension $n$ and $m$ respectively. Consider a map $f : M \to N$ and two charts $(U, ϕ)$ and $(V, ψ)$ on $M$ and $N$ respectively, so that $p \in U \subset M$ and $f (p) \in V \subset N$ , we say that $f$ is differentiable at $p$ if its coordinate representation:

Γ := ψ \circ f \circ ϕ^{- 1} : O_{U} \subset R^{n} ⟶ O_{V} \subset R^{m}, y = Γ (x) = Γ (x^{1}, . . ., x^{n})

Note: If f is onto, one-to-one and has a $C^{\infty}$ inverse then $f$ is said to be a diffeomorphism.

Vectors on a Manifold

Definition

Let $M$ be a manifold, $p \in M$ and let $F$ be the set of all $C^{\infty}$ functions from $M$ into $R$ . We define the 'tangent vector' $v$ at point $p$ to be a map $v : F \to R$ where $v$ satisfies the following two properties:

$v$ is linear i.e. $v (a f + b g) = a v (f) + b v (g)$
$v$ obeys the Leibniz rule i.e. $v (f g) = f (p) v (g) + g (p) v (f)$

The collection of all tangent vectors at $p$ is denoted by $V_{p}$ and is called the tangent space of $p$ . The space has the structure of a vector space.

Theorem

Let $M$ be an $n$ -dimensional manifold. Let $p \in M$ and let $V_{p}$ denote the tangent space at $p$ . Then dim $V_{p} = n$ .

Proof

Let $ψ : O \to U \subset R^{n}$ be a chart with $p \in O$ and let $f \in F$ . Let $X_{μ}$ be defined to be: $$ X_{\mu}(f)= \left. \frac{\partial}{\partial x^{\mu}}(f \circ \phi^{-1}) \right|{\psi(p)}$$
$X_{μ}$ gives us a basis for $V_{p}$ and is called the coordinate basis.
To show that it is indeed a basis for $V_{p}$ recall that for a real valued function $F : R^{n} \to R$ and $a \in R^{n}$ we have: $$F(x) = F(a) + \sum{\mu=1}^{n}(x^{\mu}-a^{\mu})H_{\mu}(x)$$ with $$ H_{\mu(a)}= \left.(\partial F)/(\partial x^{\mu})\right|{x=a} $$
We would apply this by letting $F = f \circ ψ^{- 1}$ and $a = ψ (p)$ then for all $q \in O$ we have $$ f(q)= f(p)+\sum\limits{\mu=1}^n [x^{\mu}\circ\psi(q) - x^{\mu}\circ\psi(p)]H_{\mu}(\psi(q))$$
Now let $v \in V_{p}$ . We would apply $f$ to the above equation. To get the following: $$ v(f) = v(f(p))+ \sum\limits_{\mu=1}^{n} \left. [x^{\mu}\circ\psi(q) - x^{\mu}\circ\psi(p)] \right|{q=p} v(H{\mu}\circ \psi) + \left.(H_{\mu}\circ\psi)\right|p v[x^{\mu}\circ\psi - x^{\mu}\circ\psi(p)] $$
Since $v (c) = 0$ for any constant $c$ . We get: $$ v(f) = \sum\limits{\mu=1}^{n}[H_{\mu}\circ\psi(p)]v(x^{\mu}\circ \psi) $$
Notice that $H_{μ} \circ ψ (p)$ is just $X_{μ} (f)$ (since we took $F = f \circ ψ^{- 1}$ ). Thus we get: $$ v(f) = \sum\limits_{\mu=1}^{n}v^{\mu}X_{\mu}(f) $$ where $v^{μ} = v (x^{μ} \circ ψ)$ (which recall is just a real number) $◻$

Frequently one denotes $X_{μ}$ as simply $\partial / \partial x^{μ}$ .