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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 ...
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.
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 ...
Given a smooth manifold M and an open real interval (a, b), a Ricci flow assigns, to each t in the interval (a,b), a Riemannian metric g t on M such that ∂ / ∂t g t = −2 Ric g t. The Ricci tensor is often thought of as an average value of the sectional curvatures , or as an algebraic trace of the Riemann curvature tensor .
A basic result of differential geometry states that normal coordinates at a point always exist on a manifold with a symmetric affine connection. In such coordinates the covariant derivative reduces to a partial derivative (at p only), and the geodesics through p are locally linear functions of t (the affine parameter).
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.