# Differential of the exponential

Let $M$ be a riemannian manifold, and $\exp_p:B_0(\epsilon)\subset T_pM \to M$ be the exponential map. How to evaluate $d(\exp_p)_vv?$

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I assume that you, in $T_pM$, identify $v$ in the origin with $v$ in $T_vT_p M$. Then it's just the tangent to the geodesic starting in $p$ with initial vector $v$ at time $t=1$. This should follow more or less directly from the definition of $\exp_p$ -- you only need a curve passing through that point for which you can calculate the tangent, and said geodesic is the best canditate.
(The question is more difficult for $d(\exp_p)_v w$ if $w\neq v$, to caluclate that you need to study Jacobi fields with initial value $w$, if I recall correctly.)
That was I did, but I was not sure... I defined $\alpha(t)=v+tv$ and compute $\frac{d\exp_p(\alpha(t))}{dt}$ in $t=0$ – Jr. Jun 10 '12 at 21:05
I'm using Do Carmo's book Riemannian Geometry, in the demonstration of Gauss lemma he assumes that you know the answer for the above question.By the way, I've alredy read the chapter of jacobi fields, so take a jacobian field with $J(0)=0$ and $J'(0)=w$ then $J(1)=d(\exp_p)_v w$ – Jr. Jun 12 '12 at 16:49