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# 53: Differential geometry

## Introduction

Differential geometry is the language of modern physics as well as an area of mathematical delight. Typically, one considers sets which are manifolds (that is, locally resemble Euclidean space) and which come equipped with a measure of distances. In particular, this includes classical studies of the curvature of curves and surfaces. Local questions both apply and help study differential equations; global questions often invoke algebraic topology.

## History

See e.g. Berger, M. "Riemannian geometry during the second half of the twentieth century", Jahresber. Deutsch. Math.-Verein. 100 (1998), no. 2, 45--208. CMP1637246

## Applications and related fields

For differential topology, See 57RXX. For foundational questions of differentiable manifolds, See 58AXX

Geometry of spheres is in the sphere FAQ. There is a separate section for detailed information about 52A55: Spherical Geometry.

A metric in the sense of differential geometry is only loosely related to the idea of a metric on a metric space.

## Subfields

• 53A: Classical differential geometry
• 53B: Local differential geometry
• 53C: Global differential geometry, see also 51H25, 58-XX; for related bundle theory, See 55RXX, 57RXX
• 53D: Symplectic geometry, contact geometry (See also 37Jxx, 70Gxx, 70Hxx) [new in 2000]

Browse all (old) classifications for this area at the AMS.

## Textbooks, reference works, and tutorials

A unique and, well, comprehensive text(s) is by Spivak, Michael: "A comprehensive introduction to differential geometry" (5 volumes) Second edition. Publish or Perish, Inc., Wilmington, Del., 1979. ISBN: 0-914098-83-7 (over 2000pp for the set!)

More comprehensible to the beginner is e.g. Barrett O'Neill's "Elementary Differential Geometry".

Intermediate: Boothby, William M. , "An introduction to differentiable manifolds and Riemannian geometry", Academic Press, Orlando, Fla, 2d edition 1986.

Harley Flanders, "Differential Forms"

Morgan, Frank: "What is a surface?", Amer. Math. Monthly 103 (1996), no. 5, 369--376. MR97h:53003

Olshanetskii , M. A.: "A short guide to modern geometry for physicists", Soviet Phys. Uspekhi 25 (1982), no. 3, 123--129 MR84a:58003

Gromov, M.: "Sign and geometric meaning of curvature", Rend. Sem. Mat. Fis. Milano 61 (1991), 9--123 (1994). MR95j:53055

Besse, Arthur L.: "Einstein manifolds", Ergebnisse der Mathematik und ihrer Grenzgebiete (3) v. 10 Springer-Verlag, Berlin-New York, 1987. 510 pp. ISBN 3-540-15279-2 MR88f:53087

## Software and tables

Ricci, A Mathematica package for doing tensor calculations in differential geometry GRG 3.2 is the computer algebra system designed for the calculations in differential geometry and field theory. The Surface Evolver, an interactive program for the study of surfaces shaped by surface tension.

## Selected topics at this site

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