Setting : Consider a two dimensional surface in $ (\mathbb{R}^n,\|\ \|)$. Here we define a function $f: \mathbb{R}^n\rightarrow \mathbb{R}^n$ s.t. $L(v)(X)=\langle f(v),X\rangle$ where $\langle\ ,\ \rangle$ is an inner product and $$L(v)(X)= \frac{d}{dt}\bigg|_{t=0}\ \frac{1}{2}\|v+tX\|^2 $$ Here note that $f(cv)=cf(v)$ for $c>0$.

Exercise : If $c$ is a path of unit speed from $p$ to $q$ in $M$, then $l(s)=\int_0^L \ \| c(t,s)_t\| \ dt$ where $c(t,s)$ is a variation of $c$. Then find a critical of a function $l$, called a geodesic

Proof : $l'(0)=\int\ L(c_t)(c_{ts})\ dt$. Here $\langle f(c_t),c_{ts}\rangle = \frac{d}{dt}\langle f(c_t),c_s\rangle - \langle df_{c_t} c_{tt},c_s\rangle$ so that $\langle df_{c_t} c_{tt},X\rangle=0$ for all $X\in T_{c(t,s)}M$ iff $c$ is a geodesic.

Remark : If $Q=df_{c_t}$, define $Q$-unit $n$ in direction $c_{tt}$, i.e. it is orthogonal to tangent space. If $ M$ is saddle, then $ \langle Qn,c_{st} \rangle^2 - \langle Q n,c_{tt} \rangle \langle Qn, c_{ss}\rangle >0$.

Question : Assume that $M$ is a saddle in $(\mathbb{R}^n,\|\ \|)$. Prove that any geodesic is minimizing.

Proof : Assume that $c(t,0)$ is a geodesic and is not minimizing between $c(0,0)$ and $c(L,0)$. If $l(s)=\int_0^L\ \|c(t,s)_t\|\ dt$, then $l''(0) <0$.

Reference : On intrinsic geometry of surface in normed spaces - Burago and Ivanv


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