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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?



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A small correction: $\mathcal{V}$ should rather be a section of $TTM$. – kangdon Sep 17 '11 at 9:30
up vote 5 down vote accepted

The higher derivatives of the exponential map satisfy the corresponding higher derivative of the Jacobi equation (because the first derivative satisfies the Jacobi equation itself), which is just an inhomogeneous Jacobi equation, where the homogeneous part is the original Jacobi equation, and the inhomogeneous term involves lower order covariant derivatives of the Jacobi field and covariant derivatives of the curvature tensor. So you would proceed recursively, bootstrapping pointwise bounds on lower derivatives, as well as pointwise bounds on the curvature tensor and its covariant derivatives, into a pointwise bound of the derivative of the Jacobi field. You'll need to figure out how get pointwise bounds for a solution to an inhomogeneous self-adjoint linear second order ODE. I'm sure this has been done before, probably for exactly the same purpose as here, but I don't know or remember where.

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Thanks Deane, I figured as much but after a couple of tentative attempts at computing it I realised I was in for some work and hoped there'd be some literature someone could point to. As you say, surely it's been done before to a similar end. I suppose it will be a good exercise. – kangdon Sep 15 '11 at 13:11
It is definitely a good exercise worth doing yourself. You should also look at papers and monographs of Jost and Karcher. For example, if you're just trying to get bounds on the metric itself with respect to "nice" co-ordinates, Jost and Karcher showed that it's better to use what they call almost-linear co-ordinates. Greene-Wu and Stefan Peters showed that these co-ordinates can be converted into harmonic co-ordinates, which have even better regularity properties. – Deane Yang Sep 15 '11 at 13:46

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.

  • Peter W. Michor: The Jacobi Flow. Rend. Sem. Mat. Univ. Pol. Torino 54, 4 (1996), 365-372 (pdf)
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