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	ABOUT: [Introduction][History][Related areas][Subfields] POINTERS: [Texts][Software][Web links][Selected topics here] 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

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

Other web sites with this focus

• Preprint server at Los Alamos
• Here are the AMS and Goettingen resource pages for area 53.
• Differential geometry for physicists
• UTK archives page

Selected topics at this site

• What are non-Euclidean Geometries?
• What is Differential Geometry; how does it differ from differential topology? May manifolds always be embedded into Euclidean space?
• What is the smallest Euclidean space into which hyperbolic k-space may be embedded?
• Is there an isometric embedding of H^2 into R^4? (open!)
• Boys' surface (pictures and formulae)
• Cyclides of Dupin (a class of surfaces in R^3)
• What are ruled, developable surfaces
• Intrinsic geometric description of parallel transport
• Constructing the geodesic joining two points on Poincare disk
• Geodesics on tori and other surfaces
• Null geodesics, Koszul formula
• Extending the Frenet vectors and formulas for curves in R^n.
• What is curvature? (N.B. -- The previous suggestion was something like: the product of the eigenvalues of the local parameterization.)
• Gaussian curvature and sectional curvature
• Gauss's Theorem Egregium: the intrinsic nature of curvature.
• Gauss's Theorema egregium (curvature is intrinsic)
• Piecewise-linear versions of Gauss-Bonnet curvature theorem
• Closed curves must have at least two points of maximal curvature.
• Minimum total curvature of a space curve
• Twist, writhe, linking numbers and applications of differential geometry to the double helix of DNA.
• How to find the closest pair of points on two circles in R^3?
• [Daniel Henry Gottlieb] Use vector fields to prove all the classical theorems! (Gauss-Bonnet, Jordan Curve, etc.)
• Nielsen fixed point theory for maps on a surface
• What are differential forms?
• DeRham's theorem links differential forms with the underlying topology of a space.
• What is foam geometrically?
• Stiefel manifold is not a symmetric space