What is riemannian metric?

Some articles on riemannian metric, metric, riemannian metrics, riemannian:

Gauss–Bonnet Theorem - Interpretation and Significance
... If the torus carries the ordinary Riemannian metric from its embedding in R3, then the inside has negative Gaussian curvature, the outside has positive Gaussian curvature, and the total ... to construct a torus by identifying opposite sides of a square, in which case the Riemannian metric on the torus is flat and has constant curvature 0, again resulting in total curvature 0 ... It is not possible to specify a Riemannian metric on the torus with everywhere positive or everywhere negative Gaussian curvature ...
Differential Geometry Of Surfaces - Local Metric Structure
... This structure is encoded infinitesimally in a Riemannian metric on the surface through line elements and area elements ... and early twentieth centuries only surfaces embedded in R3 were considered and the metric was given as a 2×2 positive definite matrix varying smoothly from point to point in a ... into account local distortions of the Earth's surface to calculate true distances, so the Riemannian metric describes distances and areas "in the small" in each ...
Differential Geometry Of Surfaces - Overview
... Smooth surfaces equipped with Riemannian metrics are of foundational importance in differential geometry ... A Riemannian metric endows a surface with notions of geodesic, distance, angle, and area ... on a closed oriented surface correspond to conformal equivalence classes of Riemannian metrics on the surface ...
Contact Geometry - Contact Forms and Structures - Relation With Symplectic Structures
... Choose a Riemannian metric on the manifold N and let H be the associated kinetic energy ... it generates the geodesic flow of the Riemannian metric ... More precisely, using the Riemannian metric, one can identify each point of the cotangent bundle of N with a point of the tangent bundle of N, and then the value of R at that point of the (unit ...
Pseudo-Riemannian Manifold
... A pseudo-Riemannian manifold is a differentiable manifold equipped with a non-degenerate, smooth, symmetric metric tensor which, unlike a Riemannian metric, need not be positive-definite, but ... Such a metric is called a pseudo-Riemannian metric and its values can be positive, negative or zero ... The signature of a pseudo-Riemannian metric is (p, q) where both p and q are non-negative ...