THE VOLUME OF TUBES IN HOMOGENEOUS SPACES

Let (')M (dim((')M) = m + n) be an oriented Riemannian manifold and M a compact oriented submanifold of (')M. The tube M(r) of radius r about M is the set of points p that can be joined to M by a geodesic of length r meeting M perpendicularly. We give a formula for the volume of M(r) in the case (')M is a naturally reductive Riemannian homogeneous space (this includes all Riemannian symmetric spaces) and M is such that for each point p of M there is a totally geodesic submanifold of (')M of dimension complementary to M through p and perpendicular to M at p.

To be more specific,

(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI)

Here h(,j) is a function of the point p (ELEM) M and the real number r. Also

h(,j)(p,r) is a homogeneous polynomial of degree j in the components

of the second fundamental form of M in (')M.