# Higher derivatives than Jacobi fields.

Hi,

the first and second derivatives of the distance function (either the full $d:M\times M\to \mathbb{R}$ function or the $d(p,\cdot):M\to \mathbb{R}$ function) as well as the derivative of the exponential map (again both of the full $\exp:TM\to M$ map and of the map $\exp_p:T_pM\to M$) may be computed with the aid of Jacobi fields, i.e, solutions to Jacobi's equation.

I have a scenario where I need second derivatives of the `full' exponential map $\exp:TM\to M$. That is, denoting the pushforward of a differentiable map by a '$_*$', I would like to know a thing or two about $\nabla_X\exp_\ast\mathcal{V}$ (where $\mathcal{V}\in TTM$ and $X$ is an appropriate vector field). In particular, I think I will require some comparison techniques analagous to those for Jacobi fields (e.g Rauch's comparison theorems).

Can anyone point me in the right direction?

Cheers,

Mat

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A small correction: $\mathcal{V}$ should rather be a section of $TTM$. –  kangdon Sep 17 '11 at 9:30

You might be interested in the Jacobi flow on $TTM$ whose flow lines project to geodesics, velocity fields of geodesics, and Jacobi fields. You can continue to higher order.