Generalized normal homogeneous Riemannian metrics on spheres and projective spaces

Berestovskiî, V. N.; Nikonorov, Yu. G.

doi:10.1007/s10455-013-9393-x

Cited by 24 publications

(30 citation statements)

References 31 publications

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“…manifolds are the generalized normal homogeneous Riemannian manifolds, also called δ-homogeneous manifolds. All metrics from this subclass are of non-negative sectional curvature (see [6,7,9]). The Clifford-Wolf homogeneous Riemannian manifolds are among the generalized normal homogeneous manifolds [8].…”

Section: Some Of Our Additional Results Includementioning

confidence: 99%

Geodesic orbit Riemannian structures on Rn

Gordon

Nikonorov

2018

Journal of Geometry and Physics

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A geodesic orbit manifold is a complete Riemannian manifold all of whose geodesics are orbits of one-parameter groups of isometries. We give both a geometric and an algebraic characterization of geodesic orbit manifolds that are diffeomorphic to R n . Along the way, we establish various structural properties of more general geodesic orbit manifolds.2010 Mathematical Subject Classification: 53C20, 53C25, 53C35.

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Section: Some Of Our Additional Results Includementioning

confidence: 99%

Geodesic orbit Riemannian structures on Rn

Gordon

Nikonorov

2018

Journal of Geometry and Physics

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“…A detailed exposition on geodesic orbit spaces and some important subclasses could be found also in [4,11,21], see also the references therein. In particular, one can find many interesting results about GO-manifolds and its subclasses in [1,2,3,6,7,8,10,12,13,18,22,23,24,27,28].…”

Section: Kvfcl On Geodesic Orbit Spacesmentioning

confidence: 99%

Spectral properties of Killing vector fields of constant length

Nikonorov

2019

Journal of Geometry and Physics

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This paper is devoted to the study of properties of Killing vector fields of constant length on Riemannian manifolds. If g is a Lie algebra of Killing vector fields on a given Riemannian manifold (M, g), and X ∈ g has constant length on (M, g), then we prove that the linear operator ad(X) : g → g has a pure imaginary spectrum. More detailed structure results on the corresponding operator ad(X) are obtained. Some special examples of vector fields of constant length are constructed.2010 Mathematical Subject Classification: 53C20, 53C25, 53C30.Key words and phrases: geodesic orbit space, homogeneous Riemannian space, Killing vector field of constant length.

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“…Many details on these Riemannian submersions can be found in papers [48], [49]. The next to the last case is the most difficult, but at the same time the most interesting case, which involves essentially the Clifford algebras Cl n and the Cayley algebra Ca of octonions.…”

Section: Proposition 2 1) There Is No Left-invariant Completely Nonhmentioning

confidence: 99%

Curvatures of homogeneous sub-Riemannian manifolds

Berestovskiî

2017

European Journal of Mathematics

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Abstract. The author proved in the late 1980s that any homogeneous manifold with an intrinsic metric is isometric to some homogeneous quotient space of a connected Lie group by its compact subgroup with an invariant Finslerian or sub-Finslerian metric. In a case of trivial compact subgroup, invariant Riemannian or sub-Riemannian metrics are singled out from invariant Finslerian or sub-Finslerian metrics by their one-to-one correspondence with special one-parameter Gaussian convolutions semigroups of absolutely continuous probability measures. Any such semigroup is generated by a second order hypoelliptic operator. In connection with this, the author discusses briefly the operator definition of Ricci lower bounds for sub-Riemannian manifolds by Baudoin-Garofalo. Earlier, Agrachev defined a notion of curvature for sub-Riemannian manifolds. As an alternative, the author discusses in some detail the old definitions of the curvature tensors for rigged metrized distributions on manifolds given by Schouten, Wagner, and Solov'ev. To calculate the Solov'ev sectional and Ricci curvatures for homogeneous sub-Riemannian manifolds, the author suggests to use in some cases special riggings of invariant completely nonholonomic distributions on manifolds. As a justification, we find a foliation on the cotangent bundle T * G over a Lie group G whose leaves are tangent to invariant Hamiltonian vector fields for the Pontryagin-Hamilton function. This function was applied in the Pontryagin maximum principle for the time-optimal problem. The foliation is entirely described by the co-adjoint representation of the Lie group G. Also we use the canonical symplectic form on T * G and its values for the above mentioned invariant Hamiltonian vector fields. In particular, the above rigging method is applicable to contact sub-Riemannian manifolds, sub-Riemannian Carnot groups, and homogeneous sub-Riemannian manifolds possessing a submetry onto a Riemannian manifold. At the end, some examples are presented.

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Generalized normal homogeneous Riemannian metrics on spheres and projective spaces

Cited by 24 publications

References 31 publications

Geodesic orbit Riemannian structures on Rn

Geodesic orbit Riemannian structures on Rn

Spectral properties of Killing vector fields of constant length

Curvatures of homogeneous sub-Riemannian manifolds

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