An Analytical Necessary and Sufficient Condition for a Riemannian Manifold to be Complete.
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.
Document Details
- Document Type
- Technical Report
- Publication Date
- Jul 06, 1972
- Accession Number
- AD0747232
Entities
People
- William B. Gordon
Organizations
- United States Naval Research Laboratory