The Volume of Tubes in Homogeneous Spaces

Citation

Howard, Ralph Elwood (1982) The Volume of Tubes in Homogeneous Spaces. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/r71m-yj10. https://resolver.caltech.edu/CaltechTHESIS:05152018-120303815

Abstract

Let M̃ (dim(M̃ ) = m + n) be an oriented Riemannian manifold and M a compact oriented submanifold of M̃ . 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 M̃ 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 M̃ of dimension complementary to M through p and perpendicular to M at p.

To be more specific,
[Equation included in scanned thesis' abstract, p. iii.]

Here h_j is a function of the point p ε 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 M̃.

Thesis Files