The Lichnerowicz–Obata Theorem on Sub-Riemannian Manifolds with Transverse Symmetries

Baudoin, Fabrice; Kim, Bumsik

doi:10.1007/s12220-014-9542-x

Cited by 19 publications

(14 citation statements)

References 22 publications

(46 reference statements)

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“…This class include the QHF, and our study has been motivated also by these works. See [22,23,25] for other comparison-type results following from the generalized CD condition.…”

Section: -Sasakian Structuresmentioning

confidence: 99%

Sub-Riemannian Ricci Curvatures and Universal Diameter Bounds for 3-Sasakian Manifolds

Rizzi

Silveira²

2017

J. Inst. Math. Jussieu

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Abstract. For a fat sub-Riemannian structure, we introduce three canonical Ricci curvatures in the sense of Agrachev-Zelenko-Li. Under appropriate bounds we prove comparison theorems for conjugate lengths, Bonnet-Myers type results and Laplacian comparison theorems for the intrinsic sub-Laplacian.As an application, we consider the sub-Riemannian structure of 3-Sasakian manifolds, for which we provide explicit curvature formulas. We prove that any complete 3-Sasakian structure of dimension 4d + 3, with d > 1, has sub-Riemannian diameter bounded by π. When d = 1, a similar statement holds under additional Ricci bounds. These results are sharp for the natural sub-Riemannian structure on S 4d+3 of the quaternionic Hopf fibrations:whose exact sub-Riemannian diameter is π, for all d ≥ 1.

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“…This class include the QHF, and our study has been motivated also by these works. See [22,23,25] for other comparison-type results following from the generalized CD condition.…”

Section: -Sasakian Structuresmentioning

confidence: 99%

Sub-Riemannian Ricci Curvatures and Universal Diameter Bounds for 3-Sasakian Manifolds

Rizzi

Silveira²

2017

J. Inst. Math. Jussieu

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“…We assume that hypotheses of Theorem are fulfilled, therefore and hold. Following the definition found in , we say that a sub‐Riemannian manifold

(M, D, g)

has transverse symmetries if

scriptV

has a basis

V_{1}, \dots, V_{m - n}

such that

{pr}_{scriptD} false[X, V_{i} false] = 0, X \in Γ false(scriptD false) .

If ∇ is the Bott connection defined in , then

\nabla_{v} V_{j} = 0 for any v \in D .

For these classes of manifolds, we state the following result.…”

Section: Transverse Symmetries and H‐type Manifoldsmentioning

confidence: 99%

“…Manifolds where is satisfied are said to be of H ‐type according to [, Section 4]. Using polarization, we get that

J_{α} J_{β} + J_{β} J_{α} = - 2 {⟨ α, β ⟩}_{g_{M}^{*}} Id .

…”

Section: Transverse Symmetries and H‐type Manifoldsmentioning

confidence: 99%

“…We assume that hypotheses of Theorem 2.4 are fulfilled, therefore (2.19) and (2.16) hold. Following the definition found in [1], we say that a sub-Riemannian manifold ( , , ) has transverse symmetries if  has a basis 1 , … , − such that pr  [ , ] = 0, ∈ Γ(). For these classes of manifolds, we state the following result.…”

Section: Transverse Symmetries and -T Ype Manifoldsmentioning

confidence: 99%

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Submersions and curves of constant geodesic curvature

Molina

Grong

Markina

2019

Mathematische Nachrichten

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Considering Riemannian submersions, we find necessary and sufficient conditions for when sub‐Riemannian normal geodesics project to curves of constant first geodesic curvature or constant first and vanishing second geodesic curvature. We describe a canonical extension of the sub‐Riemannian metric and study geometric properties of the obtained Riemannian manifold. This work contains several examples illustrating the results.

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“…In another recent paper, [19], Baudoin, Kim and Wang established a curvature-dimension inequality on Riemannian foliations with totally geodesic leaves. The interested reader might also consult the following list of references, among many others, for applications of the generalized curvature-dimension inequality: [12], [13], [14], [16], [17] and [18].…”

Section: Introductionmentioning

confidence: 99%

Sub-Riemannian Curvature of Carnot Groups with Rank-Two Distributions

Munive

2017

J Dyn Control Syst

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Abstract. The notion of curvature discussed in this paper is a far going generalization of the Riemannian sectional curvature. It was first introduced by Agrachev, Barilari and Rizzi in [2], and it is defined for a wide class of optimal control problems: a unified framework including geometric structures such as Riemannian, sub-Riemannian, Finsler, and sub-Finsler structures. In this work we study the generalized sectional curvature of Carnot groups with rank-two distributions. In particular, we consider the Cartan group and Carnot groups with horizontal distribution of Goursat-type. In these Carnot groups we characterize ample and equiregular geodesics. For Carnot groups with horizontal Goursat distribution we show that their generalized sectional curvatures depend only on the Engel part of the distribution. This family of Carnot groups contains naturally the three-dimensional Heisenberg group, as well as the Engel group. Moreover, we also show that in the Engel and Cartan groups there exist initial covectors for which there is an infinite discrete set of times at which the corresponding ample geodesics are not equiregular. Acknowledgement 1. The author wishes to thank Andrei Agrachev, Davide Barilari and Luca Rizzi for their constant interest and the many helpful conversations on sub-Riemannian geometry, especially on their paper [2]. The author also would like to thank the anonymous reviewers for their constructive comments, which helped to improve the contents of the paper, especially the content of Theorem 4.2. This work was done while the author was visiting the International School for Advanced Studies (SISSA) at Trieste, Italy.

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The Lichnerowicz–Obata Theorem on Sub-Riemannian Manifolds with Transverse Symmetries

Cited by 19 publications

References 22 publications

Sub-Riemannian Ricci Curvatures and Universal Diameter Bounds for 3-Sasakian Manifolds

Sub-Riemannian Ricci Curvatures and Universal Diameter Bounds for 3-Sasakian Manifolds

Submersions and curves of constant geodesic curvature

Sub-Riemannian Curvature of Carnot Groups with Rank-Two Distributions

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