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…)