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



Hessian formula for the sub manifold distance
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.


