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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.
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.
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 ...
In mathematics, the soul theorem is a theorem of Riemannian geometry that largely reduces the study of complete manifolds of non-negative sectional curvature to that of the compact case. Jeff Cheeger and Detlef Gromoll proved the theorem in 1972 by generalizing a 1969 result of Gromoll and Wolfgang Meyer.
In Riemannian geometry, a collapsing or collapsed manifold is an n-dimensional manifold M that admits a sequence of Riemannian metrics g i, such that as i goes to infinity the manifold is close to a k-dimensional space, where k < n, in the Gromov–Hausdorff distance sense.
Let be a complete n-dimensional Riemannian manifold whose Ricci curvature satisfies the lower bound ()for a constant .Let be the complete n-dimensional simply connected space of constant sectional curvature (and hence of constant Ricci curvature ()); thus is the n-sphere of radius / if >, or n-dimensional Euclidean space if =, or an appropriately rescaled version of n-dimensional hyperbolic ...
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.
Let (,) be a complete and smooth Riemannian manifold of dimension n. If k is a positive number with Ric g ≥ ( n -1) k , and if there exists p and q in M with d g ( p , q ) = π / √ k , then ( M , g ) is simply-connected and has constant sectional curvature k .