Gauss lemma and geodesics

From the Gauss lemma to the minimality of geodesics

Lemma:(Gauss) Suppose {p\in M}, {v\in T_u(T_pM)}

\displaystyle \langle u,v\rangle=\langle(d\exp_p)_uu,(d\exp_p)_uv\rangle

Here we make natural equivalence of {T_u(T_pM)} and {T_pM}. This lemma means {d\exp} presvers the inner product.

For any {p\in M} there exists a neighborhood {U} of {p} such that the exponential map is diffeomorphism from {B_\epsilon(0)\subset T_p(M)} to {U_p}.

Thm: {p\in M}, for such {U_p}, suppose {B=\{\exp_pv|||v||<\epsilon\}} is any geodesic ball contained in {U_p}. Then the shortest length connecting a point on {\partial B} and {p} is the geodesic ray from {p}.

Proof: Suppose {\gamma(t)=\exp_p\sigma(t):[0,1]\rightarrow M} connects {p} and {q\in \partial B}, where {\sigma(t)} is a curve in {T_pM}. Suppoe {\displaystyle \alpha(t)=\frac{\sigma(t)}{|\sigma(t)|}} is the unit normal along {\sigma(t)}, then {\dot{\sigma}(t)=\langle \dot{\sigma}(t),\alpha(t)\rangle \alpha(t)+\beta(t)} with {\beta(t)} is perpendicular to {\alpha(t)}. Then by Gauss lemma

\displaystyle \dot{\gamma}(t)=(d\exp_p)_{\sigma(t)}\dot{\sigma}(t)=(d\exp_p)_{\sigma(t)}\left[\langle\dot{\sigma}(t),\alpha(t)\rangle \alpha(t)+\beta(t)\right]

\displaystyle =\langle\dot{\sigma}(t),\alpha(t)\rangle(d\exp_p)_{\sigma(t)}\alpha(t)+(d\exp_p)_{\sigma(t)}\beta(t)

\displaystyle l(\gamma)=\int_0^1 |\dot{\gamma}(t)|dt\geq \int_0^1|\langle\dot{\sigma}(t),\alpha(t)\rangle(d\exp_p)_{\sigma(t)}\alpha(t)|dt

\displaystyle =\int_0^1 |\langle\dot{\sigma}(t),\alpha(t)\rangle|dt(\text{ since }||\alpha(t)||=1)

\displaystyle \geq \int_0^1 \langle\dot{\sigma}(t),\alpha(t)\rangle dt=\int_0^1 \langle {\sigma}(t),\alpha(t)\rangle'dt=\langle {\sigma}(1),\alpha(1)\rangle

\displaystyle =||\sigma(1)||=\text{length of }\exp t\sigma(1)= \text{geodesic ray from }p \text{ to } q

We use the fact {\langle{\sigma}(t),\dot{\alpha}(t)\rangle=0}, because {||\alpha(t)||=1} which implies {\langle{\alpha}(t),\dot{\alpha}(t)\rangle=0}.

Remark: I own this proof to Prof. Xiaochun Rong.

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