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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 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.
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 ...
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.
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.
In mathematics, complex geometry is the study of geometric structures and constructions arising out of, or described by, the complex numbers.In particular, complex geometry is concerned with the study of spaces such as complex manifolds and complex algebraic varieties, functions of several complex variables, and holomorphic constructions such as holomorphic vector bundles and coherent sheaves.
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 ...
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 ...