Riemannian Manifolds

Raymond O. Wells ()
Additional contact information
Raymond O. Wells: University of Colorado Boulder

Chapter Chapter 13 in Differential and Complex Geometry: Origins, Abstractions and Embeddings, 2017, pp 187-210 from Springer

Abstract: Abstract In 1956 Nash proved that any smooth Riemannian manifold could be isometrically embedded in a higher-dimensional Euclidean space. A fundamental tool that was developed in his paper came to be known as the Nash implicit function theorem. This theorem was a generalization of the classical implicit theorem to Banach spaces of smooth functions with specified numbers of derivatives. The differential equation that needed to be solved could be formulated as a mapping of one such Banach space to another, and near a specific type of generic embedding, the linearization of the differential equation had a right inverse. By using suitable smoothing mappings, Nash was able to generalize the classical Newton method to obtain a solution to the embedding problem near the given embedding. The general case was obtained by additional geometric analysis.

Date: 2017
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-319-58184-2_13

Ordering information: This item can be ordered from
http://www.springer.com/9783319581842

DOI: 10.1007/978-3-319-58184-2_13

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().