## Basic concepts of Riemanian Geometry

Defition of Topological manifold
Suppose ${M}$ is a topological space, for every point in ${M}$, there exists a neighborhood ${U}$ of ${x}$ and ${\phi:U\rightarrow \mathbb{R}^n}$ such that ${\psi}$ is homoemorphism of ${U}$ and an open subset in ${\mathbb{R}^n}$, then ${M}$ is called ${n-}$dimensional topological manifold.

In order to have differentiation and integration on manifold, we need ${M}$ have differentiable structure.

Suppose ${M}$ is ${n-}$dimensional manifold. If a given set of coordinate chars ${\mathcal{A}=\{(U_\alpha,\phi_\alpha)|\alpha \text{is some index}\}}$ satisfies

(1)${\bigcup\limits_\alpha U_\alpha}$ coves ${M}$

(2) the transition map between two local coordinate charts ${(U_\alpha,\phi_\alpha)}$ and ${(U_\beta,\phi_\beta)}$ are ${C^k}$ smooth. Namely the map

$\displaystyle \phi_\alpha\circ \phi_\beta^{-1}: \phi_\beta(U_\alpha\cap U_\beta)\rightarrow \phi_\alpha(U_\alpha\cap U_\beta)$

is ${C^k}$ differentiable.

(3) ${\mathcal{A}}$ is maximal. If there exists some ${(U,\phi)}$ such that it is compatible to every ${(U_\alpha,\phi_\alpha)}$, then ${(U,\phi)\subset\mathcal{A} }$.

Tangent vector on ${M}$.

A tangent vector at ${p}$ on ${M}$ is a map ${v:C_p^\infty\rightarrow \mathbb{R}^1}$ satisfies

(1) ${v(af+bg)=av(f)+b(g)}$, for ${f,g\in C^\infty_p}$ and ${a,b}$ are real numbers;

(2) ${v(fg)=v(f)g(p)+f(p)v(g)}$.

Suppose ${\gamma(t):[0,1]\rightarrow M}$, then ${\gamma}$ induces a tangent vector at ${\gamma(0)=p}$ by

$\displaystyle v(f)=\frac{d(f\circ\gamma)}{dt}\bigg|_{t=0}$

If we have a coordinate ${(u^1,u^2,\cdots,u^{n})}$ near ${p}$, then

$\displaystyle v(f)=\frac{\partial f}{\partial u^i}\frac{\partial u^i(\gamma(t))}{dt}\bigg|_{t=0}$

where ${\displaystyle \left(\frac{\partial f}{\partial u^i}\right)_p=\left(\frac{\partial f\circ \phi^{-1}}{\partial u^i}\right)_{\phi(p)}}$. Then

$\displaystyle v=\frac{\partial u^i(\gamma(t))}{dt}\bigg|_{t=0}\frac{\partial}{\partial u^i}$

is a tangent vector at ${p}$.

Tangent space

All tangent vectors form a linear space. For ${f\in C^\infty(\mathbb{R}^n)}$, there exists ${g\in C^\infty(\mathbb{R}^n)}$ such that

$\displaystyle f(u^1,u^2,\cdots,u^n)=f(0,0,\cdots,0)+ u^ig_i(u^1,u^2,\cdots,u^n)$

where ${g(0,0,\cdots,0)=\frac{\partial f}{\partial u^i}(0,0,\cdots,0)}$. Then any tangent vector ${v}$, we have

$\displaystyle v(f)=v(u^ig_i(u^1,u^2,\cdots,u^n))=\left(\frac{\partial f}{\partial u^i}\right)_pv(u^i)=\left(\frac{\partial f}{\partial u^i}\right)_p\frac{\partial}{\partial u^i}$

Moreover we can prove that ${\left\{\displaystyle \frac{\partial}{\partial u^i}\right\}_{i=0}^n}$ is linearly independent. So ${\left\{\displaystyle \frac{\partial}{\partial u^i}\right\}_{i=0}^n}$ forms a basis of the tangent space.

Tangent bundle

$\displaystyle TM=\bigcup_p T_pM=\{X_p\in T_pM|p\in M\}$

We can assign topology and differentiable stucture on tangent bundle.

Suppose ${M}$ is ${n-}$dimensional manifold. Then

$\displaystyle \pi:TM\mapsto M$

$\displaystyle X_p\rightarrow p$

For every ${X_p\in M}$, there exists a coordinates of ${p}$ on ${M}$, say ${(U,\phi)}$, then Let ${\tilde{U}=\pi^{-1}(U)=\{x\in T_pM|p\in U \}}$, define

$\displaystyle \tilde{\phi}:\tilde{U}\mapsto \phi(U)\times \mathbb{R}^n$

$\displaystyle \phi(X_p)\rightarrow (x^i(p); \xi^i)$

where ${X_p=\xi^i\frac{\partial}{\partial x^i}}$. Obviously, ${\tilde{\phi}}$ is bijective. Define the open set of ${TM}$ is ${\{\tilde{\phi}^{-1}(A\times B)|A \subset \phi(U) \text{ is open } B\subset \mathbb{R}^n\text{ is open }\}}$. So ${TM}$ is a ${2n-}$dimensional manifold, and ${\pi:TM\rightarrow M}$ is continuous.

Next suppose ${M}$ has ${C^k}$ differentiable structure. There exists an open cover ${(U_\alpha,\phi_\alpha)}$, then ${TM}$ has an open cover ${(\tilde{U}_\alpha, \tilde{\phi}_\alpha)}$ and for ${\tilde{U}_\alpha\cap \tilde{U}_\beta\neq \emptyset}$

$\displaystyle \tilde{\phi}_\alpha\circ \tilde{\phi}_\beta^{-1}: \tilde{\phi}_\beta(\tilde{U}_\alpha\cap \tilde{U}_\beta)\mapsto \tilde{\phi}_\alpha(\tilde{U}_\alpha\cap \tilde{U}_\beta)$

$\displaystyle (x_\beta^i; \xi_\beta^i)\rightarrow (x_\alpha^i; \xi_\alpha^i)$

where

$\displaystyle x_\alpha^i=(\phi_\alpha\circ\phi_\beta^{-1})^i(x_\beta^1,\cdots,x_\beta^n)$

$\displaystyle \xi_\alpha^i=\xi^k_\beta\frac{\partial (\phi_\alpha\circ\phi_\beta^{-1})^j}{\partial x_\beta^k}$

Hence ${\tilde{\phi}_\alpha\circ \tilde{\phi}_\beta^{-1}}$ is ${C^{k-1}}$ transition function. ${TM}$ has a ${C^{k-1}}$ differentiable structure.

Metric

Similarly ${T^*M}$ is the cotangent bundle which is also a manifold. Riemanian metric is a smooth function

$\displaystyle g: M\rightarrow T^*M\otimes T^*M=\bigcup_pT^*_pM\otimes T^*_pM$

satisfies

(1) ${g_{ij}(p)=g_{ji}(p)}$

(2) matrix ${(g_{ij}(p))>0}$