## derivative exponential map [closed]

let $(M,g)$ be a Riemannian manifold. and Define the exponential map by $exp : TM \rightarrow M ,exp(x,v) = \gamma(1)$ where $\gamma$ is the geodesic with $\gamma (0) = x$ and $d/dt_{t=0} \gamma = v$ and $v$ sufficiently small. My question is: what is the differential of exp in some point $exp_{\star,(x,v)} : T_{(x,v)}(TM) \rightarrow TM$ ?

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 This is discussed in many textbooks on Riemannian geometry, including Cheeger-Ebin and Gallot-Hulin-Lafontaine. It is best answered in terms of Jacobi fields. You could also ask this question on math.stackexchange.com – Deane Yang May 12 2011 at 17:32 see mathoverflow.net/questions/50712/… – Yakov Shlapentokh-Rothman May 12 2011 at 19:27