TITLE: Hessian formula for the sub manifold distance
QUESTION [6 upvotes]: 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

REPLY [3 votes]: 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.