A Formula for Popp’s Volume in Sub-Riemannian
Geometry

Barilari, Davide; Rizzi, Luca

doi:10.2478/agms-2012-0004

Cited by 79 publications

(105 citation statements)

References 16 publications

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“…For structures (not necessarily foliations) respecting the H-type condition, one can check that the Riemannian measure µ g is proportional to the intrinsic Popp's measure of the sub-Riemannian structure, and thus ∆ H coincides with the intrinsic sub-Laplacian defined with respect to Popp's measure, cf. [7].…”

Section: Sub-laplacian Comparison Theoremsmentioning

confidence: 99%

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-laplacian Comparison Theoremsmentioning

confidence: 99%

Volume and distance comparison theorems for sub-Riemannian manifolds

Baudoin

Bonnefont

Garofalo

et al. 2014

Journal of Functional Analysis

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“…In this case the Riemannian volume, denoted vol g , coincides with the canonical Popp volume of the sub-Riemannian structure (M, D, g), see [BR13]. There always exists a canonical metric and linear connection, with non-vanishing torsion Tor, called Tanno's connection.…”

Section: Applicationsmentioning

confidence: 99%

Bakry–Émery curvature and model spaces in sub-Riemannian geometry

Barilari

Rizzi

2020

Math. Ann.

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We prove comparison theorems for the sub-Riemannian distortion coefficients appearing in interpolation inequalities. These results, which are equivalent to a sub-Laplacian comparison theorem for the sub-Riemannian distance, are obtained by introducing a suitable notion of sub-Riemannian Bakry-Émery curvature. The model spaces for comparison are variational problems coming from optimal control theory. As an application we establish the sharp measure contraction property for 3-Sasakian manifolds satisfying a suitable curvature bound.

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“…See [Mon02,BR13] for the definition of the Popp measure and Example 3.20 for the case of step-2 Carnot groups. In these cases, we denote the corresponding Jacobians as J Haus f and J Popp f , respectively.…”

Section: Equivalence Of Metric Definitionsmentioning

confidence: 99%

Conformality and Q-harmonicity in sub-Riemannian manifolds

Capogna

Citti

Donne

et al. 2019

Journal de Mathématiques Pures et Appliquées

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Abstract. We prove the equivalence of several natural notions of conformal maps between sub-Riemannian manifolds. Our main contribution is in the setting of those manifolds that support a suitable regularity theory for subelliptic p-Laplacian operators. For such manifolds we prove a Liouville-type theorem, i.e., 1-quasiconformal maps are smooth. In particular, we prove that contact manifolds support the suitable regularity. The main new technical tools are a sub-Riemannian version of p-harmonic coordinates and a technique of propagation of regularity from horizontal layers.

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A Formula for Popp’s Volume in Sub-Riemannian Geometry

Cited by 79 publications

References 16 publications

Volume and distance comparison theorems for sub-Riemannian manifolds

Volume and distance comparison theorems for sub-Riemannian manifolds

Bakry–Émery curvature and model spaces in sub-Riemannian geometry

Conformality and Q-harmonicity in sub-Riemannian manifolds

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