53: Differential geometry
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.
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.
Browse all (old) classifications for this area at the AMS.
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