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A Sasakian metric is defined using the construction of the Riemannian cone.Given a Riemannian manifold (,), its Riemannian cone is the product (>)of with a half-line >, equipped with the cone metric
In mathematics, a Riemannian manifold is said to be flat if its Riemann curvature tensor is everywhere zero. Intuitively, a flat manifold is one that "locally looks like" Euclidean space in terms of distances and angles, e.g. the interior angles of a triangle add up to 180°.
A manifold is asymptotically simple if it admits a conformal compactification ~ such that every null geodesic in has future and past endpoints on the boundary of ~.. Since the latter excludes black holes, one defines a weakly asymptotically simple manifold as a manifold with an open set isometric to a neighbourhood of the boundary of ~, where ~ is the conformal compactification of some ...
A closely related question is the so-called "non-compact Yamabe problem", which asks: Is it true that on every smooth complete Riemannian manifold (M,g) which is not compact, there exists a metric that is conformal to g, has constant scalar curvature and is also complete?
Formally, a Riemannian metric (or just a metric) on a smooth manifold is a choice of inner product for each tangent space of the manifold. 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 ...
Metric units are units based on the metre, gram or second and decimal (power of ten) multiples or sub-multiples of these. According to Schadow and McDonald, [ 1 ] metric units, in general, are those units "defined 'in the spirit' of the metric system, that emerged in late 18th century France and was rapidly adopted by scientists and engineers.
is also a Riemannian metric on . We say that ~ is (pointwise) conformal to . Evidently, conformality of metrics is an equivalence relation. Here are some formulas for conformal changes in tensors associated with the metric.
Divergences are generally notated with an uppercase 'D', as in (,), to distinguish them from metric distances, which are notated with a lowercase 'd'. When multiple divergences are in use, they are commonly distinguished with subscripts, as in D KL {\displaystyle D_{\text{KL}}} for Kullback–Leibler divergence (KL divergence).