Riemannian manifold


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Riemannian manifold

[rē′män·ē·ən ′man·ə‚fōld]
(mathematics)
A differentiable manifold where the tangent vectors about each point have an inner product so defined as to allow a generalized study of distance and orthogonality.
References in periodicals archive ?
In mathematics this ambient space is a Riemannian manifold, i.e.
The components of quasi-conformal like curvature tensor W in a Riemannian manifold ([M.sup.2n+1], g)(n > 1), are given by
All SPD matrices form a Riemannian manifold S++ when endowed with a Riemannian metric.
The method we use for computing the energy of Bishop vector fields in this study is that considering a vector field as a map from manifold M to the Riemannian manifold (TM, [p.sub.s]), where TM is tangent bundle of a Riemannian manifold and [p.sub.s] is a Sasaki metric induced from TM naturally.
General relativity (GR) [1] formulates a physical problem in terms of differential equations as a geometric requirement that a space-time may correspond to a Riemannian manifold as the interaction of matter and gravitation.
The notion of Golden structure on a Riemannian manifold was introduced for the first time by C.E.
A manifold endowed with a smoothly varying inner product in tangent spaces is called a Riemannian manifold. Next, we introduce the definition of the gradient [2].
It is well known that any complete, simply connected Riemannian manifold having nonpositive sectional curvature is a CAT(0) space.
If the form [psi] is closed, M is a partially bi-Kahlerian submanifolds, i.e., a Riemannian manifold such that its metric [psi] has two de Rham decompositions that have one Kaahlerian term [5].
Qin, Volume growth and escape rate of Brownian motion on a complete Riemannian manifold, Ann.
In [5], the energy of a unit vector field X on a Riemannian manifold M is defined as the energy of the mapping X : M [right arrow] [T.sup.1] M, where the unit tangent bundle [T.sup.1] M is equipped with the restriction of the Sasaki metric on TM.
To sum up, a d-dimensional orientable Riemannian manifold admits a globally defined volume form which leads to the isomorphism between [[OMEGA].sup.k] (M) and [[OMEGA].sup.d-k] (M).