Chern, S. S. Formerly, Mathematical Sciences Research Institute, University of California, Berkeley, California.
- Riemannian spaces
- Covariant differentiation
- Parallelism and geodesics
- Riemann-Christoffel tensor
- Manifolds and bundles
- Space-forms and symmetric spaces
- Gauss-Bonnet formula
- DeRham and Hodge theorems
- Finslerian geometry
- Symplectic geometry
- Complex manifolds
- Additional Readings
The geometry of Riemannian manifolds. Riemannian geometry was initiated by B. Riemann in 1854, following the pioneering work of C. F. Gauss on surface theory in 1827. Riemann introduced a coordinate space in which the infinitesimal distance between two neighboring points is specified by a quadratic differential form, given below. Such a space is a natural generalization of Euclidean geometry and Gauss's geometry of surfaces in three-dimensional euclidean space, as well as the non-Euclidean geometries: hyperbolic geometry (previously discovered by J. Bolyai and N. I. Lobachevsky) and elliptic geometry. A Riemannian manifold is a topological space that further generalizes this notion. Riemannian geometry derives great importance from its application in the general theory of relativity, where the universe is considered to be a Riemannian manifold. See also: Differential geometry; Euclidean geometry; Non-Euclidean geometry; Relativity
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