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A Riemannian manifold is a smooth manifold together with a Riemannian metric. The techniques of differential and integral calculus are used to pull geometric data out of the Riemannian metric. For example, integration leads to the Riemannian distance function, whereas differentiation is used to define curvature and parallel transport.
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 ...
Download as PDF; Printable version; ... Pages in category "Riemannian manifolds" The following 41 pages are in this category, out of 41 total. ... Simons' formula ...
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.
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 mathematical field of Riemannian geometry, the Reilly formula is an important identity, discovered by Robert Reilly in 1977. [1] It says that, given a smooth Riemannian manifold-with-boundary ( M , g ) and a smooth function u on M , one has
In Riemannian geometry, the smooth coarea formulas relate integrals over the domain of certain mappings with integrals over their codomains. Let M , N {\displaystyle \scriptstyle M,\,N} be smooth Riemannian manifolds of respective dimensions m ≥ n {\displaystyle \scriptstyle m\,\geq \,n} .
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 .