Ok, recall our goal was to prove Helgason’s formula, $\displaystyle \boxed{ (d \exp)_{tX}(Y) = \left( \frac{ 1 - e^{\theta( - tX^* )}}{\theta(tX^*)} (Y^*) \right)_{\exp(tX)}.}$

and that we have already shown $\displaystyle {(d \exp)_{tX}(Y) f = \sum_{n=0}^{\infty} \frac{t^n}{(n+1)!} ( X^{*n} Y^* + X^{*(n-1)} Y^* X^* + \dots + Y^* X^{*n})f(p).}$  (more…)

We showed that the differential of the exponential map ${\exp_p: T_p(M) \rightarrow M}$ for ${M}$ a smooth manifold and ${p \in M}$ is the identity at ${0 \in T_p(M)}$. In the case of analytic manifolds, it is possible to say somewhat more. First of all, if we’re working with real-analytic manifolds, we can say that a connection ${\nabla}$ is analytic if ${\nabla_XY}$ is analytic for analytic vector fields ${X,Y}$. Using the real-analytic versions of the ODE theorem, it follows that ${\exp_p}$ is an analytic morphism.

So, make the above assumptions: analyticity of both the manifold and the connection. Now there is a small disk ${V_p \subset T_p(M)}$ such that ${\exp_p}$ maps ${V_p}$ diffeomorphically onto a neighborhood ${U \subset M}$ containing ${p}$. We will compute ${d(\exp_p)_{X}(Y)}$ when ${X \in V_p}$ is sufficiently small and ${Y \in T_p(M)}$ (recall that we identify ${T_p(M)}$ with its tangent spaces at each point). (more…)