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I am working on some problems involving foliations and group actions and would be very nice to consider the second derivatives for the distance function of an orbit or a leaf.

So my question is: does anyone know a reference or an easy way to compute the hessian tensor associated to the function:

$d : M \to \mathbb{R}$, where $d(p) := \mathrm{dist}(p,L),$ where $L$ is a Riemannian sub manifold of the Riemannian manifold $M$.

Further, in the case $L$ is an orbit (possibly singular) of a Lie group acting on $M$, does the formula become easier to compute?

I ve done some research and on Petersen's book the only formulae I could find was about distance from a point, not a general sub manifold.

Thank you very much

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    $\begingroup$ The classic reference for this is the paper by Heintze and Karcher. $\endgroup$ – Deane Yang Feb 12 at 16:49
  • $\begingroup$ @DeaneYang, do you mean "A general comparison theorem with applications to volume estimates for submanifolds"? Could not find any related to this there. $\endgroup$ – L.F. Cavenaghi Feb 12 at 17:14
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    $\begingroup$ The hessian of the distance function is the second fundamental form of the hypersurfaces at constant distances from your submanifold. This satisfies a Riccati equation, and therefore can be controlled from above and below by comparison techniques once the ambient curvature is bounded. Heintze and Karcher should indeed be a good reference for this. $\endgroup$ – Raziel Feb 12 at 17:32
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    $\begingroup$ @L.F.Cavenaghi, Raziel is correct, but you probably can't find everything described in detail in one place. I believe Petersen is a good place to read about Jacobi fields, the fact that the second fundamental form is the Hessian of the distance function, and how to use the comparison theory of the matrix Riccati equation to obtain bounds on the Hessian. Heintze-Karcher extend the approach used for distance from a point or a hypersurface to the distance from a submanifold. $\endgroup$ – Deane Yang Feb 12 at 19:08
  • $\begingroup$ thank you very much to all the answers, I think I can solve it now. $\endgroup$ – L.F. Cavenaghi Feb 13 at 0:27
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I believe an answer to your question can be found in the book:

A. Gray, Tubes. Second edition. Progress in Mathematics, 221. Birkhäuser Verlag, Basel, 2004.

If you know how to search, you can find this book online.

The title Tubes refers to a tubular neighborhoods of radius $r$ of a submanifold. One of the point is to find a formula for the volume of a tube. Quite surprisingly, it is possible to find an exact formula in terms related to curvatures. Computing volume requires a lot of computations with the distance function so I would suggest to look into that book and I hope you will find there what you are looking for.

The results of E. Heintze and H. Karcher mentioned in comments to your question are also discussed in that book.

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