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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 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 differential geometry, the Laplace–Beltrami operator is a generalization of the Laplace operator to functions defined on submanifolds in Euclidean space and, even more generally, on Riemannian and pseudo-Riemannian manifolds. It is named after Pierre-Simon Laplace and Eugenio Beltrami.
Such a solution is called a harmonic function and such solutions are the topic of study in potential theory. In a more general setting, where Ω ⊆ R n is replaced by any Riemannian manifold M, and u : Ω → R is replaced by u : M → Φ for another (different) Riemannian manifold Φ, the Dirichlet energy is given by the sigma model.
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 existence of isothermal coordinates on a smooth two-dimensional Riemannian manifold is a corollary of the standard local solvability result in the analysis of elliptic partial differential equations. In the present context, the relevant elliptic equation is the condition for a function to be harmonic relative to
In mathematics, specifically in the field of differential topology, Morse homology is a homology theory defined for any smooth manifold.It is constructed using the smooth structure and an auxiliary metric on the manifold, but turns out to be topologically invariant, and is in fact isomorphic to singular homology.
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 ...