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He proved the existence of Green's function on arbitrary simply connected domains other than itself; this established the Riemann mapping theorem. [3] Constantin Carathéodory gave another proof of the theorem in 1912, which was the first to rely purely on the methods of function theory rather than potential theory. [4]
In Riemannian geometry and pseudo-Riemannian geometry, the Gauss–Codazzi equations (also called the Gauss–Codazzi–Weingarten-Mainardi equations or Gauss–Peterson–Codazzi formulas [1]) are fundamental formulas that link together the induced metric and second fundamental form of a submanifold of (or immersion into) a Riemannian or pseudo-Riemannian manifold.
The technical statement appearing in Nash's original paper is as follows: if M is a given m-dimensional Riemannian manifold (analytic or of class C k, 3 ≤ k ≤ ∞), then there exists a number n (with n ≤ m(3m+11)/2 if M is a compact manifold, and with n ≤ m(m+1)(3m+11)/2 if M is a non-compact manifold) and an isometric embedding ƒ: M → R n (also analytic or of class C k). [15]
The Cartan–Hadamard theorem in conventional Riemannian geometry asserts that the universal covering space of a connected complete Riemannian manifold of non-positive sectional curvature is diffeomorphic to R n. In fact, for complete manifolds of non-positive curvature, the exponential map based at any point of the manifold is a covering map.
For a surface M embedded in E 3 (or more generally a higher-dimensional Euclidean space), there are several equivalent definitions of a vector field X on M: a smooth map of M into E 3 taking values in the tangent space at each point; the velocity vector of a local flow on M; a first order differential operator without constant term in any local ...
In Riemannian geometry, Gauss's lemma asserts that any sufficiently small sphere centered at a point in a Riemannian manifold is perpendicular to every geodesic through the point. More formally, let M be a Riemannian manifold, equipped with its Levi-Civita connection, and p a point of M. The exponential map is a mapping from the tangent space ...
Composition of (r,φ) with the inverse of the exponential map at p is a polar coordinate system. Polar coordinates provide a number of fundamental tools in Riemannian geometry. The radial coordinate is the most significant: geometrically it represents the geodesic distance to p of nearby points.
There exist non-geodesically complete compact pseudo-Riemannian (but not Riemannian) manifolds. An example of this is the Clifton–Pohl torus . In the theory of general relativity , which describes gravity in terms of a pseudo-Riemannian geometry, many important examples of geodesically incomplete spaces arise, e.g. non-rotating uncharged ...