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Do Carmo's main research interests were Riemannian geometry and the differential geometry of surfaces. [3]In particular, he worked on rigidity and convexity of isometric immersions, [26] [27] stability of hypersurfaces [28] [29] and of minimal surfaces, [30] [31] topology of manifolds, [32] isoperimetric problems, [33] minimal submanifolds of a sphere, [34] [35] and manifolds of constant mean ...
The dot products on every tangent plane, packaged together into one mathematical object, are a Riemannian metric. 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.
The existence of isothermal coordinates can be proved [6] by applying known existence theorems for the Beltrami equation, which rely on L p estimates for singular integral operators of Calderón and Zygmund. [7] [8] A simpler approach to the Beltrami equation has been given more recently by Adrien Douady. [9] If the Riemannian metric is given ...
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.
The fundamental theorem of Riemannian geometry states that on any Riemannian manifold (or pseudo-Riemannian manifold) there is a unique affine connection that is torsion-free and metric-compatible, called the Levi-Civita connection or (pseudo-) Riemannian connection of the given metric.
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.
In the case of a Riemannian 2-manifold, the fundamental theorem of Riemannian geometry can be rephrased in terms of Cartan's canonical 1-forms: Theorem. On an oriented Riemannian 2-manifold M , there is a unique connection ω on the frame bundle satisfying
Hopf–Rinow theorem is a set of statements about the geodesic completeness of Riemannian manifolds. It is named after Heinz Hopf and his student Willi Rinow , who published it in 1931. [ 1 ] Stefan Cohn-Vossen extended part of the Hopf–Rinow theorem to the context of certain types of metric spaces .