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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 ...
The geodesic flow of any compact Riemannian manifold with negative sectional curvature is ergodic. If M is a complete Riemannian manifold with sectional curvature bounded above by a strictly negative constant k then it is a CAT space. Consequently, its fundamental group Γ = π 1 (M) is Gromov hyperbolic. This has many implications for the ...
The class ManifoldOpenSubset has been suppressed: open subsets of manifolds are now instances of TopologicalManifold or DifferentiableManifold (since an open subset of a top/diff manifold is a top/diff manifold by itself) Functions defined on a coordinate patch are no longer necessarily symbolic functions of the coordinates: they now pertain to ...
A Riemannian manifold is a differentiable manifold in which each tangent space is equipped with an inner product , in a manner which varies smoothly from point to point. Given two tangent vectors u {\displaystyle u} and v {\displaystyle v} , the inner product u , v {\displaystyle \langle u,v\rangle } gives a real number.
The Ricci Flow in Riemannian Geometry: A Complete Proof of the Differentiable 1/4-Pinching Sphere Theorem (1st ed.). Springer. ISBN 978-3-642-16285-5. Jost, Jürgen (2005). Riemannian geometry and Geometric Analysis (4th ed.). Springer. ISBN 978-3-540-25907-7. Lee, Jeffrey M. (2009). Manifolds and Differential Geometry. American Mathematical ...
Differentiable functions between two manifolds are needed in order to formulate suitable notions of submanifolds, and other related concepts. If f : M → N is a differentiable function from a differentiable manifold M of dimension m to another differentiable manifold N of dimension n, then the differential of f is a mapping df : TM → TN.
The Riemannian connection or Levi-Civita connection [9] is perhaps most easily understood in terms of lifting vector fields, considered as first order differential operators acting on functions on the manifold, to differential operators on sections of the frame bundle. In the case of an embedded surface, this lift is very simply described in ...