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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.
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 ...
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
Since any Ricci flow on a two-dimensional manifold is confined to a single conformal class, it can be recast as a partial differential equation for a scalar function on the fixed Riemannian manifold (M, g 0). As such, the Ricci flow in this setting can also be studied by purely analytic methods; correspondingly, there are alternative non ...
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.
Just as there are various types of manifolds, there are various types of maps of manifolds. PDIFF serves to relate DIFF and PL, and it is equivalent to PL.. In geometric topology, the basic types of maps correspond to various categories of manifolds: DIFF for smooth functions between differentiable manifolds, PL for piecewise linear functions between piecewise linear manifolds, and TOP for ...
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.
If is a Lie group and a Riemannian manifold with a faithful action of by isometries then the action is analytic. Usually one takes to be the full isometry group of .Then the category of (,) manifolds is equivalent to the category of Riemannian manifolds which are locally isometric to (i.e. every point has a neighbourhood isometric to an open subset of ).