The Subelliptic Heat Kernels of the Quaternionic Hopf Fibration

Baudoin, Fabrice; Wang, Jing

doi:10.1007/s11118-014-9403-z

Cited by 28 publications

(43 citation statements)

References 25 publications

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“…We point out that points (a) and (b) are sharp, as the upper diameter bound is attained for the case of the standard sub-Riemannian structure on the complex, quaternionic or octonionic Hopf fibrations (see e.g. [19,20,14], respectively).…”

Section: Sub-riemannian Diameter Boundsmentioning

confidence: 92%

Volume and distance comparison theorems for sub-Riemannian manifolds

Baudoin

Bonnefont

Garofalo

et al. 2014

Journal of Functional Analysis

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On H-type sub-Riemannian manifolds we establish sub-Hessian and sub-Laplacian comparison theorems which are uniform for a family of approximating Riemannian metrics converging to the sub-Riemannian one. We also prove a sharp sub-Riemannian Bonnet-Myers theorem that extends to this general setting results previously proved on contact and quaternionic contact manifolds.The above formulas show that the Hladky connection defined relative to g ε in (2.1) will coincide for all choices of ε > 0.Let (M, g) be a Riemannian manifold, equipped with an orthogonal splitting T M = H ⊕ V. If V is integrable and metric, then (M, H, g) is a Riemannian foliation with bundlelike metric and totally geodesic leaves, tangent to V. We simply refer to these structures as totally geodesic foliations. In this foliation context, Hladky connection is referred to as the Bott connection (see [17]). The H-type and the J 2 conditionsWe introduce a condition that will play a prominent role in the following. For any Z ∈ T M, let J Z : T M → T M be defined asWe remark that J is defined for any Riemannian manifold (M, g) equipped with an orthogonal splitting T M = H ⊕ V.Remark 2.3. If (M, g) is a totally geodesic foliation, it holdsDefinition 2.4. We say that the H-type condition is satisfied ifDefinition 2.5 ([26, 22]). We say that J 2 condition holds if for all Z, Z ′ ∈ Γ(V), X ∈ Γ(H) with Z, Z ′ = 0 there exists Z ′′ ∈ Γ(V) such that

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Section: Sub-riemannian Diameter Boundsmentioning

confidence: 92%

Volume and distance comparison theorems for sub-Riemannian manifolds

Baudoin

Bonnefont

Garofalo

et al. 2014

Journal of Functional Analysis

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“…However, in the complex and quaternionic case the Lie group structure of the fiber played an important role that we can not use here, since the fiber S 7 is not a group. Instead, we make use of some algebraic properties of S 7 that were already pointed out and used by the authors in [1] for the study of the octonionic Hopf fibration:…”

Section: Introduction and Resultsmentioning

confidence: 99%

Integral representation of the sub-elliptic heat kernel on the complex anti-de Sitter fibration

Baudoin

Демни

2018

Arch. Math.

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“…Proof. The identities (13) follow directly from the definitions while (14) is precisely (12) written in terms of A, C and v. For (15), working in the Fermi frame along γ, we havė…”

Section: Lemma 16 the Non-zero Brackets Between ∂mentioning

confidence: 99%

A Bonnet–Myers type theorem for quaternionic contact structures

Barilari

Ivanov

2019

Calc. Var.

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We prove a Bonnet-Myers type theorem for quaternionic contact manifolds of dimension bigger than 7. If the manifold is complete with respect to the natural sub-Riemannian distance and satisfies a natural Ricci-type bound expressed in terms of derivatives up to the third order of the fundamental tensors, then the manifold is compact and we give a sharp bound on its sub-Riemannian diameter.for some (and then any) set X 1 , . . . , X k ∈ Γ(D) of local generators for D.Given a sub-Riemannian structure on M , the sub-Riemannian distance is defined by:d SR (x, y) = inf{ℓ(γ) | γ(0) = x, γ(T ) = y, γ horizontal}.

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The Subelliptic Heat Kernels of the Quaternionic Hopf Fibration

Cited by 28 publications

References 25 publications

Volume and distance comparison theorems for sub-Riemannian manifolds

Volume and distance comparison theorems for sub-Riemannian manifolds

Integral representation of the sub-elliptic heat kernel on the complex anti-de Sitter fibration

A Bonnet–Myers type theorem for quaternionic contact structures

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