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Riemannian Manifolds

In: Differential and Complex Geometry: Origins, Abstractions and Embeddings

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  • Raymond O. Wells Jr.

    (University of Colorado Boulder
    Jacobs University Bremen)

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.

Suggested Citation

  • Raymond O. Wells Jr., 2017. "Riemannian Manifolds," Springer Books, in: Differential and Complex Geometry: Origins, Abstractions and Embeddings, chapter 0, pages 187-210, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-58184-2_13
    DOI: 10.1007/978-3-319-58184-2_13
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