Transverse Weitzenböck formulas and curvature dimension inequalities on Riemannian foliations with totally geodesic leaves

Fabrice Baudoin; Bumsik Kim; Jing Wang

doi:10.4310/CAG.2016.v24.n5.a1

Transverse Weitzenböck formulas and curvature dimension inequalities on Riemannian foliations with totally geodesic leaves

Fabrice Baudoin, Bumsik Kim, Jing Wang

Publikation: Bidrag til tidsskrift/Konferencebidrag i tidsskrift /Bidrag til avis › Tidsskriftartikel › Forskning › peer review

18 Citationer (Scopus)

Abstract

We prove a family of Weitzenböck formulas on a Riemannian foliation with totally geodesic leaves. These Weitzenböck formulas are naturally parametrized by the canonical variation of the metric. As a consequence, under natural geometric conditions, the horizontal Laplacian satisfies a generalized curvature dimension inequality. Among other things, this curvature dimension inequality implies Li-Yau estimates for positive solutions of the horizontal heat equation, sharp eigenvalue estimates and a sub-Riemannian Bonnet- Myers compactness theorem whose assumptions only rely on the intrinsic geometry of the horizontal distribution.

Originalsprog	Engelsk
Tidsskrift	Communications in Analysis and Geometry
Vol/bind	24
Nummer	5
Sider (fra-til)	913-937
Antal sider	25
ISSN	1019-8385
DOI	https://doi.org/10.4310/CAG.2016.v24.n5.a1
Status	Udgivet - 2016
Udgivet eksternt	Ja

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title = "Transverse Weitzenb{\"o}ck formulas and curvature dimension inequalities on Riemannian foliations with totally geodesic leaves",

abstract = "We prove a family of Weitzenb{\"o}ck formulas on a Riemannian foliation with totally geodesic leaves. These Weitzenb{\"o}ck formulas are naturally parametrized by the canonical variation of the metric. As a consequence, under natural geometric conditions, the horizontal Laplacian satisfies a generalized curvature dimension inequality. Among other things, this curvature dimension inequality implies Li-Yau estimates for positive solutions of the horizontal heat equation, sharp eigenvalue estimates and a sub-Riemannian Bonnet- Myers compactness theorem whose assumptions only rely on the intrinsic geometry of the horizontal distribution.",

keywords = "Bochner method, Riemannian foliation, Sub-Riemannian geometry, Weitzenb{\"o}ck formula",

author = "Fabrice Baudoin and Bumsik Kim and Jing Wang",

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doi = "10.4310/CAG.2016.v24.n5.a1",

language = "English",

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pages = "913--937",

journal = "Communications in Analysis and Geometry",

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Transverse Weitzenböck formulas and curvature dimension inequalities on Riemannian foliations with totally geodesic leaves. / Baudoin, Fabrice; Kim, Bumsik; Wang, Jing.
I: Communications in Analysis and Geometry, Bind 24, Nr. 5, 2016, s. 913-937.

Publikation: Bidrag til tidsskrift/Konferencebidrag i tidsskrift /Bidrag til avis › Tidsskriftartikel › Forskning › peer review

TY - JOUR

T1 - Transverse Weitzenböck formulas and curvature dimension inequalities on Riemannian foliations with totally geodesic leaves

AU - Baudoin, Fabrice

AU - Kim, Bumsik

AU - Wang, Jing

PY - 2016

Y1 - 2016

N2 - We prove a family of Weitzenböck formulas on a Riemannian foliation with totally geodesic leaves. These Weitzenböck formulas are naturally parametrized by the canonical variation of the metric. As a consequence, under natural geometric conditions, the horizontal Laplacian satisfies a generalized curvature dimension inequality. Among other things, this curvature dimension inequality implies Li-Yau estimates for positive solutions of the horizontal heat equation, sharp eigenvalue estimates and a sub-Riemannian Bonnet- Myers compactness theorem whose assumptions only rely on the intrinsic geometry of the horizontal distribution.

AB - We prove a family of Weitzenböck formulas on a Riemannian foliation with totally geodesic leaves. These Weitzenböck formulas are naturally parametrized by the canonical variation of the metric. As a consequence, under natural geometric conditions, the horizontal Laplacian satisfies a generalized curvature dimension inequality. Among other things, this curvature dimension inequality implies Li-Yau estimates for positive solutions of the horizontal heat equation, sharp eigenvalue estimates and a sub-Riemannian Bonnet- Myers compactness theorem whose assumptions only rely on the intrinsic geometry of the horizontal distribution.

KW - Bochner method

KW - Riemannian foliation

KW - Sub-Riemannian geometry

KW - Weitzenböck formula

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U2 - 10.4310/CAG.2016.v24.n5.a1

DO - 10.4310/CAG.2016.v24.n5.a1

M3 - Journal article

AN - SCOPUS:85015928972

SN - 1019-8385

VL - 24

SP - 913

EP - 937

JO - Communications in Analysis and Geometry

JF - Communications in Analysis and Geometry

IS - 5

ER -

Transverse Weitzenböck formulas and curvature dimension inequalities on Riemannian foliations with totally geodesic leaves

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