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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 Riemannian connection or Levi-Civita connection [9] is perhaps most easily understood in terms of lifting vector fields, considered as first order differential operators acting on functions on the manifold, to differential operators on sections of the frame bundle. In the case of an embedded surface, this lift is very simply described in ...
The connection Laplacian, also known as the rough Laplacian, is a differential operator acting on the various tensor bundles of a manifold, defined in terms of a Riemannian- or pseudo-Riemannian metric. When applied to functions (i.e. tensors of rank 0), the connection Laplacian is often called the Laplace–Beltrami operator.
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.
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 ...
The notion of a differentiable manifold refines that of a manifold by requiring the functions that transform between charts to be differentiable. In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus.
On an oriented manifold, 1-densities can be canonically identified with the n-forms on M. On non-orientable manifolds this identification cannot be made, since the density bundle is the tensor product of the orientation bundle of M and the n-th exterior product bundle of T ∗ M (see pseudotensor).