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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.
In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, [1] is a theorem relating the flux of a vector field through a closed surface to the divergence of the field in the volume enclosed.
The divergence of a vector field extends naturally to any differentiable manifold of dimension n that has a volume form (or density) μ, e.g. a Riemannian or Lorentzian manifold. Generalising the construction of a two-form for a vector field on R 3, on such a manifold a vector field X defines an (n − 1)-form j = i X μ obtained by contracting ...
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 ...
If a compact Riemannian n-manifold has non-negative Ricci curvature, then its first Betti number is at most n, with equality if and only if the Riemannian manifold is a flat torus. Splitting theorem. If a complete n-dimensional Riemannian manifold has nonnegative Ricci curvature and a straight line (i.e. a geodesic that minimizes distance on ...
The Laplace–Beltrami operator, like the Laplacian, is the (Riemannian) divergence of the (Riemannian) gradient: = (). An explicit formula in local coordinates is possible. Suppose first that M is an oriented Riemannian manifold.
An extension of the fundamental theorem states that given a pseudo-Riemannian manifold there is a unique connection preserving the metric tensor, with any given vector-valued 2-form as its torsion. The difference between an arbitrary connection (with torsion) and the corresponding Levi-Civita connection is the contorsion tensor .
If is a compactly supported vector field and is a manifold with boundary, then Stokes' theorem implies () = ⌋, which is a generalization of the divergence theorem. The solenoidal vector fields are those with div X = 0. {\displaystyle \operatorname {div} X=0.}