# Second variation in saddle Finsler surface

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