On Clifford-Wolf homogeneous Riemannian manifolds

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

doi:10.1134/s106456240806001x

Cited by 30 publications

(98 citation statements)

References 35 publications

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“…where∇ is the Levi-Civita connection ofĝ (see [4], p. 474, Proposition 1). Thus by Proposition 2.3 we have K (P, y) =K (P) ≥ 0.…”

Section: Theorem 51 a Restrictively Cw-homogeneous Finsler Space Hasmentioning

confidence: 99%

“…Recall that simply connected CW-homogeneous Riemannian manifolds have been classified by Berestovskii and Nikonorov in [4]. The list consists of the Euclidean spaces, the odd-dimensional spheres of constant curvature, compact simple Lie groups with bi-invariant Riemannian metrics, and the Riemannian product of the above three types of manifolds.…”

Section: Introductionmentioning

confidence: 99%

“…Berestovskii and Yu.G. Nikonorov in the Riemannian case; see [2][3][4]. We refer to [21][22][23] for more information on the study of CW-translations on Riemannian manifolds.…”

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confidence: 99%

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$$(\alpha _1,\alpha _2)$$ ( α 1 , α 2 ) -Metrics and Clifford–Wolf Homogeneity

Deng

2015

J Geom Anal

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In this paper, we introduce new types of Finsler metrics, called (α 1 , α 2 )-metrics. We define the notion of the good datum of a homogeneous (α 1 , α 2 )-metric and use that to study the geometric properties. In particular, we give a formula for the S-curvature and deduce a condition for the S-curvature to be vanishing identically. Moreover, we consider the restrictive Clifford-Wolf homogeneity of left invariant (α 1 , α 2 )-metrics on compact connected simple Lie groups. We prove that, in some special cases, a restrictively Clifford-Wolf homogeneous (α 1 , α 2 )-metric must be Riemannian. An unexpected interesting observation contained in the proof reveals the fact that S-curvature may play an important role in the study of Clifford-Wolf homogeneity in Finsler geometry. on spheres have been classified recently by the authors in [27]. Therefore our next step is to classify left invariant CW-homogeneous Finsler metrics on compact Lie groups. In our previous works, we have performed this program for Randers metrics and (α, β)-metrics (see [12,25]). Therefore we study CW-homogeneity of left invariant (α 1 , α 2 )-metrics on compact Lie groups in this paper. However, the general case seems to be very involved, so we will confine ourselves to the case where in the decomposition g = V 1 + V 2 of T G e = g, the subspace V 2 is a commutative subalgebra of g. In particular, we will discuss the following two cases:Case 1 G is a compact connected simple Lie group, and V 2 is a Cartan subalgebra. Case 2 G is a compact connected simple Lie group, and V 2 is 2-dimensional commutative subalgebra.In the study of the restrictive CW-homogeneity of left invariant non-Riemannian (α 1 , α 2 )-metrics in Case 1, the S-curvature plays an important role. The main results are the following two theorems.The study of the restrictive CW-homogeneity of left invariant non-Riemannian (α 1 , α 2 )-metrics of Case 2 is a generalization of our work on (α, β)-metrics [25]. The systematic technique we have developed in the study of Killing vector fields of constant length of left invariant Randers and (α, β)-metrics also works in this case. (α 1 , α 2 )-metric on a compact connected simple Lie group G with a decomposition g = V 1 + V 2 such that V 2 is a 2-dimensional commutative subalgebra. If F is restrictively CW-homogeneous, then it must be Riemannian. Theorem 1.2 Let F be a left invariantTheorems 1.1 and 1.2 suggest the following conjecture: Conjectute 1.3 Let F be a left invariant (α 1 , α 2 )-metric on a compact connected simple Lie group G with a decomposition g = V 1 + V 2 such that V 2 is a commutative subalgebra. If F is restrictively CW-homogeneous, then it must be Riemannian.More generally, we can make a stronger one: (α 1 , α 2 )-metric on a compact connected simple Lie group G with a dimension decomposition (n 1 , n 2 ), where n 1 ≥ n 2 > 1. If F is restrictively CW-homogeneous, then it must be Riemannian. Conjectute 1.4 Let F be a left invariantThe paper is organized as follows. In Sect. 2, we recall some definitions and known res...

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“…where∇ is the Levi-Civita connection ofĝ (see [4], p. 474, Proposition 1). Thus by Proposition 2.3 we have K (P, y) =K (P) ≥ 0.…”

Section: Theorem 51 a Restrictively Cw-homogeneous Finsler Space Hasmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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$$(\alpha _1,\alpha _2)$$ ( α 1 , α 2 ) -Metrics and Clifford–Wolf Homogeneity

Deng

2015

J Geom Anal

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“…In [36], it is proven, Lemma 3, and in [37] is used, that a Killing vector field X ∈ (M) has constant length if and only if every integral curve of X is a geodesic, that is, X is a geodesic vector field. We can give a little more general proof:…”

Section: Geodesic Fields and Sundman Transformationmentioning

confidence: 99%

Infinitesimal Time Reparametrisation and Its Applications

Cariñena

Martínez

Muñoz-Lecanda³

2022

J Nonlinear Math Phys

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“…An interesting subclass of δ-homogeneous manifolds is that of Clifford-Wolf homogeneous manifolds, defined as those δ-homogeneous manifolds for which equality holds in (1.1) for all x ∈ M . The simply connected Clifford-Wolf manifolds were classified in [3] by using properties of Killing fields of constant length.…”

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confidence: 99%