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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 ...
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.
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.
In Riemannian geometry, it is a traditional name for a number of theorems that compare various metrics and provide various estimates in Riemannian geometry. [4] Rauch comparison theorem relates the sectional curvature of a Riemannian manifold to the rate at which its geodesics spread apart; Toponogov's theorem; Myers's theorem; Hessian ...
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 ...
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 ...
Let (M, g) denote a Riemannian manifold, and consider an oriented smooth manifold S (possibly with boundary) together with a one-parameter family of smooth immersions f t of S into M. For each individual value of the parameter t , the immersion f t induces a Riemannian metric on S , which itself induces a differential form on S known as the ...
Distributions satisfying a further integrability condition give rise to foliations, i.e. partitions of the manifold into smaller submanifolds. These notions have several applications in many fields of mathematics, including integrable systems, Poisson geometry, non-commutative geometry, sub-Riemannian geometry, differential topology.