Diferential of exponential map

Question

Let $\exp: (p,w) \in TM \to \exp_{p}w = \gamma_{p,w}(1) \in M$ where $w \in T_{p}M$. We have that

$d(\exp)_{(p,w)}: T_{(p,w)}TM \to T_{\gamma_{p,w}(1)}M$.

Given $\xi \in T_{(p,w)}TM$. How we calculate $d(\exp)_{(p,w)}(\xi)$?

Att

Answer 1

It is the value of a Jacobi field along $\gamma_{(p,w)}$. More precisely, there is a unique Jacobi field $J$ along $\gamma_{(p,w)}$ that satisfies $J(0)=0$ and $D_tJ(0)= \xi$, and then $d(exp)_{(p,w)}(\xi) = J(1)$.

You can't explicitly compute this vector except in very special cases such as constant-curvature metrics. But if you have bounds on the sectional curvature, you can often obtain upper or lower bounds on its size. Some of this is explained in my Riemannian Manifolds, Chapters 10 and 11. (More will be explained in the second edition, which hopefully will be out next year.)

For more detailed info, you can consult other Riemannian geometry books such as Petersen or do Carmo.