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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.
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 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.
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
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 regular homotopy between two immersions f and g from a manifold M to a manifold N is defined to be a differentiable function H : M × [0,1] → N such that for all t in [0, 1] the function H t : M → N defined by H t (x) = H(x, t) for all x ∈ M is an immersion, with H 0 = f, H 1 = g. A regular homotopy is thus a homotopy through immersions.