Minimizing properties of geodesics problem in do Carmo's book

Question

I'm DoCarmo's book Riemannian Geometry and in the section with minimizing properties of geodesics it this proposition. I don't understand why $\langle\frac{\partial f} {\partial r}, \frac{\partial f} {\partial t} \rangle=0$. Can someone fill in the details?

This is the Gauss lemma that he is talking about. So my question becomes, how did he apply this lemma in order to obtain that inner product equal to zero.

Answer 1

Note that $f(r,t) = \exp(rv(t))$, hence by the chain rule$$ \partial_r f(r_0,t_0)=(d\exp_p)_{r_0v(t_0)}[v(t_0)] $$ and $$ \partial _t f(r_0,t_0)=(d\exp_p)_{r_0v(t_0)}[r_0\dot v(t_0)]. $$ Hence $$ \langle \partial_r f(r_0,t_0)\vert \partial_t f(r_0,t_0)\rangle = \langle (d\exp_p)_{r_0v(t_0)}[v(t_0)]~\vert~ (d\exp_p)_{r_0v(t_0)}[r_0\dot v(t_0)]\rangle\\ =r_0^{-1}\langle (d\exp_p)_{r_0v(t_0)}[r_0v(t_0)]~\vert~ (d\exp_p)_{r_0v(t_0)}[r_0\dot v(t_0)]\rangle \overset{\text{Gauß}}{=} r_0^{-1} \langle r_0 v(t_0) \vert r_0\dot v(t_0) \rangle. $$ The latter is zero as it is a multiple to the derivative of $t\mapsto \langle v(t)\vert v(t) \rangle \equiv 1$.