Let $(M,g)$ be a complete Riemannian manifold. Suppose $\gamma_1,\gamma_2:[0,1] \to M$ are two different segments(i.e.minimizing geodesics) from $p$ to $x=\exp_p(v)$. Suppose further that $\gamma'_1(1)+\gamma'_2(1)\neq 0$, and thus we can find some $w\in T_xM$ such that $g(w,\gamma'_i(1))< 0$ (Intuitively,$w$ forms an angle $> 90^{\circ}$ with each $\gamma'_i(1)$, $i=1,2$ ).
Now, we select a curve $c:[0,\epsilon) \to M$ with $c(0)=x$ and $c'(0)=w$. I would like to ask how to show that $d(p,c(s))<d(p,x)$ for any $0<s<\epsilon$.
Please help. Thanks!
