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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.
Sylvestre Gallot, IHES, Bures-sur-Yvette 2007 Sylvestre F. L. Gallot (born January 29, 1948, in Bazoches-lès-Bray ) [ 1 ] [ 2 ] is a French mathematician, specializing in differential geometry . He is an emeritus professor at the Institut Fourier of the Université Grenoble Alpes , in the Geometry and Topology section.
Let be a smooth manifold and let be a one-parameter family of Riemannian or pseudo-Riemannian metrics. Suppose that it is a differentiable family in the sense that for any smooth coordinate chart, the derivatives v i j = ∂ ∂ t ( ( g t ) i j ) {\displaystyle v_{ij}={\frac {\partial }{\partial t}}{\big (}(g_{t})_{ij}{\big )}} exist and are ...
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.
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
Riemannian Geometry (PDF). Princeton: Princeton University Press. OCLC 5836010. Eisenhart, Luther Pfahler (1939). Coordinate Geometry. Dover Publishing. [7] Eisenhart, Luther Pfahler (1927). Non-Riemannian geometry (PDF). New York: American Mathematical Society. [8] Eisenhart, Luther Pfahler (1909). A treatise on the differential geometry of ...
Cartan connection. Cartan-Hadamard space is a complete, simply-connected, non-positively curved Riemannian manifold.. Cartan–Hadamard theorem is the statement that a connected, simply connected complete Riemannian manifold with non-positive sectional curvature is diffeomorphic to R n via the exponential map; for metric spaces, the statement that a connected, simply connected complete ...
In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds.The sectional curvature K(σ p) depends on a two-dimensional linear subspace σ p of the tangent space at a point p of the manifold.