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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.
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 ...
On a Riemannian manifold it is an elliptic operator, while on a Lorentzian manifold it is hyperbolic. The Laplace–de Rham operator is defined by = + = (+), where d is the exterior derivative or differential and δ is the codifferential, acting as (−1) kn+n+1 ∗d∗ on k-forms, where ∗ is the Hodge star.
A basic example of maps between manifolds are scalar-valued functions on a manifold, : or :, sometimes called regular functions or functionals, by analogy with algebraic geometry or linear algebra. These are of interest both in their own right, and to study the underlying manifold.
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.
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 ...
In the mathematical field of differential geometry, a smooth map between Riemannian manifolds is called harmonic if its coordinate representatives satisfy a certain nonlinear partial differential equation. This partial differential equation for a mapping also arises as the Euler-Lagrange equation of a functional called the Dirichlet energy.
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.