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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 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.
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 conjecture has a more general form, sometimes called the "generalized Cartan–Hadamard conjecture" [12] which states that if the curvature of the ambient Cartan–Hadamard manifold M is bounded above by a nonpositive constant k, then the least perimeter enclosures in M, for any given volume, cannot have smaller perimeter than a sphere enclosing the same volume in the model space of ...
The Levi-Civita connection is named after Tullio Levi-Civita, although originally "discovered" by Elwin Bruno Christoffel.Levi-Civita, [1] along with Gregorio Ricci-Curbastro, used Christoffel's symbols [2] to define the notion of parallel transport and explore the relationship of parallel transport with the curvature, thus developing the modern notion of holonomy.
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). This gives, in particular, local notions of angle, length of curves, surface area and volume.
Klingenberg's area of work was geometry, especially differential geometry and Riemannian geometry.Besides many articles he published several books. One of his major achievements was the proof of the sphere theorem in joint work with Marcel Berger in 1960: The sphere theorem states that a complete, simply connected Riemannian manifold with sectional curvature contained in the interval (1, 4] is ...
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