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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.)

Edit (adding a reference): This fact is also known as the Gauss Lemma, a reference would be Lemma 1.9.1 in Klingenbergs 'Riemannian Geometry', 2nd edition, de Gruyter. The remark about Jacobi fields is discussed in chapter 1.12 of that book.

As a further note, this fact is needed to prove the standard so called comparison theorems in Riemannian geometry, so any textbook adressing these will discuss this topic.

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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
    
@Jr. I added a reference where you can look up a detailed proof. –  user20266 Jun 12 '12 at 16:02
    
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
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