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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 ...
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.
Let (M,g) be a closed smooth Riemannian manifold. Then there exists a positive and smooth function f on M such that the Riemannian metric fg has constant scalar curvature. By computing a formula for how the scalar curvature of fg relates to that of g, this statement can be rephrased in the following form: Let (M,g) be
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.
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
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 ...
Using this, we can consider an SDE on the orthonormal frame bundle of a Riemannian manifold, whose solution is Brownian motion, and projects down to the (base) manifold via stochastic development. A visual representation of this construction corresponds to the construction of a spherical Brownian motion by rolling without slipping the manifold ...