Accession Number:

AD0747232

Title:

An Analytical Necessary and Sufficient Condition for a Riemannian Manifold to be Complete.

Descriptive Note:

Final rept.,

Corporate Author:

NAVAL RESEARCH LAB WASHINGTON D C MATHEMATICS RESEARCH CENTER

Personal Author(s):

Report Date:

1972-07-06

Pagination or Media Count:

5.0

Abstract:

It is known that every differentiable manifold supports a complete Riemannian structure. This is a consequency of Whitneys Embedding Theorem. Moreover, Nomizu and Ozeki have shown that every Riemannian manifold is conformally equivalent to a complete Riemannian manifold. In this report a method is given for constructing complete Riemannian metrics which is exceedingly simple and provides a necessary and sufficient condition for the completeness of a Riemannian structure. Namely, it is phonon that a Riemannian manifold is complete if and only if it supports a proper function whose gradient is bounded in modulus.

Subject Categories:

  • Theoretical Mathematics

Distribution Statement:

APPROVED FOR PUBLIC RELEASE