Search results
Results from the WOW.Com Content Network
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 Hopf manifolds are examples of complex manifolds that are not Kähler. To construct one, take a complex vector space minus the origin and consider the action of the group of integers on this space by multiplication by exp(n). The quotient is a complex manifold whose first Betti number is one, so by the Hodge theory, it cannot be Kähler.
Complex projective space, CP n; Quaternionic projective space, HP n; Flag manifold; Grassmann manifold; Stiefel manifold; Lie groups provide several interesting families. See Table of Lie groups for examples. See also: List of simple Lie groups and List of Lie group topics.
Riemannian manifolds and Riemann surfaces are named after Bernhard Riemann. In 1857, Riemann introduced the concept of Riemann surfaces as part of a study of the process of analytic continuation; Riemann surfaces are now recognized as one-dimensional complex manifolds. He also furthered the study of abelian and other multi-variable complex ...
A Kähler manifold is a Riemannian manifold of even dimension whose holonomy group is contained in the unitary group (). [3] Equivalently, there is a complex structure on the tangent space of at each point (that is, a real linear map from to itself with =) such that preserves the metric (meaning that (,) = (,)) and is preserved by parallel transport.
There are several equivalent definitions of a Riemann surface. A Riemann surface X is a connected complex manifold of complex dimension one. This means that X is a connected Hausdorff space that is endowed with an atlas of charts to the open unit disk of the complex plane: for every point x ∈ X there is a neighbourhood of x that is homeomorphic to the open unit disk of the complex plane, and ...
Any 3-Sasakian manifold is both an Einstein manifold and a spin manifold. If M is positive-scalar-curvature Kähler–Einstein manifold, then, by an observation of Shoshichi Kobayashi , the circle bundle S in its canonical line bundle admits a Sasaki–Einstein metric, in a manner that makes the projection from S to M into a Riemannian submersion.
Marcel Berger's 1955 paper [2] on the classification of Riemannian holonomy groups first raised the issue of the existence of non-symmetric manifolds with holonomy Sp(n)·Sp(1).Interesting results were proved in the mid-1960s in pioneering work by Edmond Bonan [3] and Kraines [4] who have independently proven that any such manifold admits a parallel 4-form .The long awaited analog of strong ...