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  2. Riemann mapping theorem - Wikipedia

    en.wikipedia.org/wiki/Riemann_mapping_theorem

    This mapping is known as a Riemann mapping. [1] Intuitively, the condition that be simply connected means that does not contain any “holes”. The fact that is biholomorphic implies that it is a conformal map and therefore angle-preserving. Such a map may be interpreted as preserving the shape of any sufficiently small figure, while possibly ...

  3. Gauss–Codazzi equations - Wikipedia

    en.wikipedia.org/wiki/Gauss–Codazzi_equations

    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.

  4. Myers's theorem - Wikipedia

    en.wikipedia.org/wiki/Myers's_theorem

    Since is connected, there exists the smooth universal covering map :. One may consider the pull-back metric π * g on N . {\displaystyle N.} Since π {\displaystyle \pi } is a local isometry, Myers' theorem applies to the Riemannian manifold ( N ,π * g ) and hence N {\displaystyle N} is compact and the covering map is finite.

  5. Exponential map - Wikipedia

    en.wikipedia.org/wiki/Exponential_map

    exponential map (Riemannian geometry) for a manifold with a Riemannian metric, exponential map (Lie theory) from a Lie algebra to a Lie group, More generally, in a manifold with an affine connection, (), where is a geodesic with initial velocity X, is sometimes also called the exponential map. The above two are special cases of this with ...

  6. Nash embedding theorems - Wikipedia

    en.wikipedia.org/wiki/Nash_embedding_theorems

    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]

  7. Exponential map (Riemannian geometry) - Wikipedia

    en.wikipedia.org/wiki/Exponential_map...

    The exponential map of the Earth as viewed from the north pole is the polar azimuthal equidistant projection in cartography. In Riemannian geometry, an exponential map is a map from a subset of a tangent space T p M of a Riemannian manifold (or pseudo-Riemannian manifold) M to M itself. The (pseudo) Riemannian metric determines a canonical ...

  8. Complete manifold - Wikipedia

    en.wikipedia.org/wiki/Complete_manifold

    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 ...

  9. Riemannian manifold - Wikipedia

    en.wikipedia.org/wiki/Riemannian_manifold

    Riemannian manifolds are named after German mathematician Bernhard Riemann, who first conceptualized them. Formally, a Riemannian metric (or just a metric) on a smooth manifold is a choice of inner product for each tangent space of the manifold. A Riemannian manifold is a smooth manifold together with a Riemannian metric.

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