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) : Gordon,William B.

Report Date : 06 JUL 1972

Pagination or Media Count : 5

Abstract : It is known that every differentiable manifold supports a complete Riemannian structure. This is a consequency of Whitney's 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.

Descriptors :   (*ALGEBRAIC GEOMETRY, THEOREMS), (*DIFFERENTIAL GEOMETRY, TENSOR ANALYSIS), SET THEORY, OPERATORS(MATHEMATICS), GEODESICS

Subject Categories : Theoretical Mathematics

Distribution Statement : APPROVED FOR PUBLIC RELEASE