Riemannian in a sentence

Any oriented Riemannian (or pseudo-Riemannian ) manifold has a natural volume (or pseudo volume) form.

By contrast with Riemannian geometry, where the curvature provides a local invariant of Riemannian manifolds, Darboux's theorem states that all symplectic manifolds are locally isomorphic.

Explicitly, for any two tangent vectors : we have : Analogously, isometric immersion is an immersion between Riemannian manifolds which preserves the Riemannian metrics.

An exact replacement of Whitehead's cosmology would need to admit a Riemannian geometry.

A partition of unity can be used to show the existence of a Riemannian metric on an arbitrary manifold.

A Whiteheadian process is most importantly characterized by extension in space-time, marked by a continuum of uncountably many points in a Minkowski or a Riemannian space-time.

Complex structures on a closed oriented surface correspond to conformal equivalence classes of Riemannian metrics on the surface.

Differential forms and the differential can be defined on any Euclidean space, or indeed any manifold, without any notion of a Riemannian metric.

Geometric proofs There exists still another way to approach the fundamental theorem of algebra, due to J. M. Almira and A. Romero: by Riemannian Geometric arguments.

Many concepts and techniques of analysis and differential equations have been generalized to the setting of Riemannian manifolds.

Minkowski space first approximates the universe without gravity ; the pseudo-Riemannian manifolds of general relativity describe spacetime with matter and gravity.

Over a compact subset of M, this follows by fixing a Riemannian metric on M and using the exponential map for that metric.

Put differently, a Riemannian manifold is a space constructed by deforming and patching together Euclidean spaces.

Ricci flow with surgery main Hamilton's program for proving the Poincaré conjecture involves first putting a Riemannian metric on the unknown simply connected closed 3-manifold.

Riemannian geometry generalizes Euclidean geometry to spaces that are not necessarily flat, although they still resemble the Euclidean space at each point infinitesimally, i.e. in the first order of approximation.

Riemannian manifolds are special cases of the more general Finsler manifolds.

Riemannian manifolds The atomic orbitals of chemistry are spherical harmonics and can be used to produce Fourier series on the sphere main If the domain is not a group, then there is no intrinsically defined convolution.

The device knew a renewed interest as the Tonnetz in neo-Riemannian theory (see also Lattice (music) ).

The introduction of a Riemannian metric or a symplectic form gives rise to a natural isomorphism between the tangent space and the cotangent space at a point, associating to any tangent covector a canonical tangent vector.

The metric tensor that defines the geometry—in particular, how lengths and angles are measured—is not the Minkowski metric of special relativity, it is a generalization known as a semi- or pseudo-Riemannian metric.