# Classical Surface Theory, the Theorema Egregium of Gauss, and Differential Geometry on Manifolds

Chapter

## Abstract

In this and the following two chapters we consider three central applications of the theory of manifolds:

- (i)
Classical surface theory of Gauss.

- (ii)
Riemannian and affine connected manifolds.

- (iii)
Einstein’s general theory of relativity (1916).

## Keywords

Riemannian Manifold Fundamental Form Tensor Field Parallel Transport Christoffel Symbol## Preview

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## References to the Literature

- Classical works: Euclid (325 B.C.) (“Elements”), Gauss (1827) (surface theory), Riemann (1854) (Riemannian manifolds), Beltrami (1868) (construction of a two- dimensional Riemannian manifold with negative curvature, with non-Euclidean geometry), Klein (1871) (models for non-Euclidean geometries in the context of projective geometry), Klein (1872) (Erlanger program-group-theoretical classification of the geometries), Hilbert (1903, M) (axiomatic foundation for the general geometry), Ricci and Levi-Civita (1901) (covariant differentiation), Einstein (1916) (applications of the calculus of covariant differentiation to the general theory of relativity), Levi-Civita (1917) (parallel transport).Google Scholar
- History of non-Euclidean geometry and the concept of manifolds: Klein (1926, M), (1928, M), Scholz (1980, M).Google Scholar
- Collected works which contain important contributions in the development of geometry: Gauss (1863), Riemann (1892), Klein (1921), Poincaré (1928), Hilbert (1932), Lie (1934), E. Cartan (1952), Einstein (1960).Google Scholar
- Gauss biographies: Worbs (1955, M), Wussing (1974, M), Bühler (1981, M).Google Scholar
- Classical surface theory: Kreyszig (1957, M, B, R), Laugwitz (1960, M).Google Scholar
- Surface deformation in the large: Efimov (1957, M).Google Scholar
- Introduction to classical tensor calculus: Schouten (1954, M) (standard work), Raschewski (1959, M), Zeidler (1979, S) (connection between vector analysis, tensor analysis, differential forms, and differential geometry).Google Scholar
- Classical differential geometry: Blaschke (1923, M), (1950, M, H) (applications of differential forms).Google Scholar
- Modern differential geometry: Spivak (1979, M, H, B), Vols. 1–5 (recommended as a comprehensive introduction), Helgason (1962, M), Kobayashi and Nomizu (1963, M), Sternberg (1964, M), Sulanke and Wintgen (1972, M), Greub, Halperin, and Vanstone (1972, M), Choquet-Bruhat (1982, M).Google Scholar
- Riemannian geometry: Klingenberg (1983, M), Besse (1987, M).Google Scholar
- Isometric embedding of Riemannian manifolds in the ℝ
^{n}: Nash (1956) (fundamental work), Schwartz (1969, L), Gromov and Rohlin (1970, S, B), Gromov (1986, M).Google Scholar - Non-Euclidean geometry: Klein (1928, M).Google Scholar
- Theorem of Gauss-Bonnet-Chern. Classical works: Gauss (1827), Bonnet (1848), Dyck (1885), Chern (1944). Introduction: Kreyszig (1957, M), Guillemin and Pollack (1974, M), Sulanke and Wintgen (1972, M). Connection with the theory of characteristic classes: Spivak (1979, M), Vol. 5 (recommended as an introduction), Chern (1959, L), Schwartz (1968, L), Greub, Halperin, and Vanstone (1972, M), Vol. 2 (general theory).Google Scholar
- Differential forms: Maurin (1976, M), Vol. 2 and Westenholz (1981, M) (introduction), Kähler (1934, M) (applications to systems of partial differential equations), Hodge (1952, M), Cartan (1955, M), de Rahm (1960, M), Greub, Halperin, and Vanstone (1972, M), Vols. 1-3, Marsden (1974, L), Abraham and Marsden (1978, M) (applications to mechanics).Google Scholar
- Lie groups: Choquet-Bruhat (1982) (introduction), Chevalley (1946, M), Montgomery and Zippin (1955, M), Pontrjagin (1966, M), Warner (1974, M), Dieudonne (1975, M), Vols. 4 and 5.Google Scholar
- Modern differential geometry and its applications to mathematical physics: Dubrovin, Novikov, and Fomenko (1979, M), Westenholz (1981, M), Choquet-Bruhat (1982, M), Curtis (1986, M).Google Scholar

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© Springer Science+Business Media New York 1988