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.
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A Riemannian metric g is said to be associated with a contact manifold if there exist a (1, 1) tensor ield [phi] and a contravariant global vector field [xi], called the characteristic vector field of the manifold such that
gives a positive definite Riemannian metric called the lift metric.
In effect, the space-time is splittable into 3 + 1 dimensions and the induced Riemannian metric [g.sub.ij] onto the three-dimensional [[summation].sub.t] can be used to write the Laplace-Beltrami operator [[DELTA].sub.LB], which is the kinetic energy term in the Schrodinger equation for the subjected to the geometric field quantum particle or quantum condensate:
Regardless of the values of the functions a, b [member of] [C.sup.[infinity]]([OMEGA]), the Riemannian metric given by (7) satisfies that [pi] is a Killing submersion over ([OMEGA], [ds.sup.2.sub.[lambda]]) with [xi] = [[partial derivative].sub.z] as unit vertical Killing field.
The induced Riemannian metric g on Mis given by [bar.g](X, Y) = [bar.g]([i.sub.*]X, [i.sub.*]Y), for any X, Y [member of] [GAMMA](TM), where [GAMMA](TM) denotes the set of all vector fields of M.
Let [M.sup.2n] be a generalized almost Hermitian manifold, with the generalized Riemannian metric G and the compatible generalized almost complex structure J.
spaces, in Section 4 we discuss how to find all homogeneous geodesics for a given G-invariant Riemannian metric.
Let [S.sup.n] [subset] [R.sup.n+1] be the unit sphere, and let us identify the tangent and cotangent bundles of [S.sup.n.sub.1] by means of the standard Riemannian metric. The unit (co)sphere bundle of [S.sup.n.sub.1] is given by the incidence correspondence
On the frame bundle, a Riemannian metric on spacetime manifold M is replaced by a local orthonormal basis [E.sub.A] (A = 1, ..., d) of the tangent bundle TM.
Integrating this we obtain [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [bar.[gamma]] is a fixed Riemannian metric on the initial slice (t = [t.sub.0] a constant).
Reference [7] designed a Riemannian metric on point cloud to reflect the surface for finding the most desirable direction (as flat as possible) for traversal with reduced energy consumption.
MEller, Peloso, and Ricci consider the Hodge Laplacian delta on the Heisenberg group endowed with a left-variant and U(n)-invariant Riemannian metric. The topics include differential forms and the Hodge Laplacian on Hn; Bargmann representations and sections of homogeneous bundles; core, domains, and self-adjount extensions, first properties of deltak exact and closed forms; intertwining operators and different scalar forms for deltak; unitary intertwining operators and projections, Lp-multipliers; and applications.