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The Volume of Tubes in Homogeneous Spaces


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


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 hj is a function of the point p ε M and the real number r. Also hj(p,r) is a homogeneous polynomial of degree j in the components of the second fundamental form of M in M̃.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Mathematics
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Mathematics
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Conn, Jack Frederick
Thesis Committee:
  • Conn, Jack Frederick (chair)
  • Aschbacher, Michael
  • De Prima, Charles R.
  • Fuller, F. Brock
Defense Date:7 May 1982
Funding AgencyGrant Number
Record Number:CaltechTHESIS:05152018-120303815
Persistent URL:
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:10890
Deposited By: Mel Ray
Deposited On:25 Jun 2018 21:30
Last Modified:14 Jun 2023 23:06

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