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}

 

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