Riemannian geometry - Wikipedia
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as smooth manifolds with a Riemannian metric (an inner product on the tangent space at each point that varies smoothly from point to point).
The study of Riemannian Geometry is rather meaningless without some basic knowledge on Gaussian Geometry that is the di erential geometry of curves and surfaces in 3-dimensional space.
Riemannian geometry | Differential geometry, Manifolds ...
Riemannian geometry, one of the non-Euclidean geometries that completely rejects the validity of Euclid’s fifth postulate and modifies his second postulate. Simply stated, Euclid’s fifth postulate is: through a point not on a given line there is only one line parallel to the given line.
Riemannian manifold - Wikipedia
In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the -sphere, hyperbolic space, and smooth surfaces in three-dimensional space, such as ellipsoids and paraboloids, are all examples of Riemannian manifolds.
Riemannian Geometry - SpringerLink
Intended for a one year course, this text serves as a single source, introducing readers to the important techniques and theorems, while also containing enough background on advanced topics to appeal to those students wishing to specialize in Riemannian geometry.
Riemannian geometry - Encyclopedia of Mathematics
2011年2月7日 · Riemannian geometry is a multi-dimensional generalization of the intrinsic geometry (cf. Interior geometry) of two-dimensional surfaces in the Euclidean space . The metric of a Riemannian space coincides with the Euclidean metric of the domain under consideration up to the first order of smallness.
2.1 Riemannian Metrics Definition 2.1Let Mbe a smooth manifold. A symmetric tensor fieldΦ on Mof type (0,2) is a map Φ : X(M) ×X(M) −→C∞(M) satisfying: (i) Φ(X,Y) = Φ(Y,X), (ii) Φ(X,fY+ gZ) = fΦ(X,Y) + gΦ(X,Z), for all X,Y,Z∈X(M) and f,g∈C∞(M). Definition 2.2Let Mbe a smooth manifold. A Riemannian metric gis a symmetric positive