Category Archives: Research

Handbook of Discrete and Computational Geometry (3rd Edition, 2016)

Handbook of Discrete and Computational Geometry
—Third Edition—
edited by Jacob E. Goodman, Joseph O’Rourke, and Csaba D. Tóth
CRC Press LLC, to appear (2016).

http://www.csun.edu/~ctoth/Handbook/HDCG3.html
Previous editions:

  1. Handbook of Discrete and Computational Geometry, First Edition
    J.E. Goodman and J. O’Rourke, editors,
    CRC Press LLC, Boca Raton, FL, 1997.
    ISBN 978-0849385247 (52 chapters, xiv + 991 pages).
  2. Handbook of Discrete and Computational Geometry, Second Edition
    J.E. Goodman and J. O’Rourke, editors,
    CRC Press LLC, Boca Raton, FL, 2004.
    ISBN 978-1584883012 (65 chapters, xvii + 1539 pages).

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Notes on the Nash embedding theorem

Terence Tao notes on embedding theorem.

What's new

Throughout this post we shall always work in the smooth category, thus all manifolds, maps, coordinate charts, and functions are assumed to be smooth unless explicitly stated otherwise.

A (real) manifold $latex {M}&fg=000000$ can be defined in at least two ways. On one hand, one can define the manifold extrinsically, as a subset of some standard space such as a Euclidean space $latex {{bf R}^d}&fg=000000$. On the other hand, one can define the manifold intrinsically, as a topological space equipped with an atlas of coordinate charts. The fundamental embedding theorems show that, under reasonable assumptions, the intrinsic and extrinsic approaches give the same classes of manifolds (up to isomorphism in various categories). For instance, we have the following (special case of) the Whitney embedding theorem:

Theorem 1 (Whitney embedding theorem) Let $latex {M}&fg=000000$ be a compact manifold. Then there exists an embedding $latex {u: M rightarrow {bf…

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