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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.
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.
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.
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 conformal manifold is a Riemannian manifold (or pseudo-Riemannian manifold) equipped with an equivalence class of metric tensors, in which two metrics g and h are equivalent if and only if =, where λ is a real-valued smooth function defined on the manifold and is called the conformal factor.
Related: 300 Trivia Questions and Answers to Jumpstart Your Fun Game Night What Is Today's Strands Hint for the Theme: "Thar She Blows!"? Today's Strands game revolves around marine mammals.
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 ...