Jacobi fields / conjugate points problem in do Carmo's book

Question

I'm reading do Carmo's book, riemannian geometry and I don't know how to prove this remark. Can some one fill in some of the details??

Answer 1

The Jacobi equation is a second-order linear ODE. Hence, by standard ODE theory, the space of solutions (i.e. Jacobi fields) is linear with dimension $2n$; for every $u,v\in T_{\gamma(0)}M$ there exists a unique Jacobi field $J$ satisfying $$J(0)=u,\;\left.\frac{D}{dt}\right|_{t=0}J(t)=v.$$ Fixing $u=0$ leaves us with an $n$-dimensional space where solutions are determined by $v$.

The fact that $J(t)=t\dot{\gamma}(t)$ is a Jacobi field can be verified by a straightforward calculation. It certainly vanishes at $t=0$, and it does not vanish anywhere else as $|\dot{\gamma}(t)|$ is constant in $t$.