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Generalized Ricci curvature bounds for three dimensional contact subriemannian manifolds

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Abstract

Measure contraction property is one of the possible generalizations of Ricci curvature bound to more general metric measure spaces. In this paper, we discover necessary and sufficient conditions for a three dimensional contact subriemannian manifold to satisfy this property.

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Abbreviations

\(M\) :

A metric space or a manifold

\(\langle \cdot ,\cdot \rangle ,\; \left| \cdot \right| \) :

A subriemannian metric and its norm

\(\Delta \) :

A distribution on \(M\)

\(\eta \) :

The Popp’s measure

\(\mu ,\; \mu _0,\mu _1,\; \mu _t\) :

Measures on \(M\)

\(\Pi \) :

A measure on \(M\times M\)

\(d\) :

A distance function on \(M\)

\(U\) :

A Borel set in \(M\)

\(H\) :

Subriemannian Hamiltonian

\(e^{t\vec H}\) :

Subriemannian geodesic flow

\(e_i(t),f_i(t)\) :

Canonical Darboux frame

\(R^{ij}_\alpha (t)\) :

Curvature invariants

\(v\) :

Tangent vectors

\(\alpha \) :

Covectors

\(\alpha _0\) :

Contact form

\(v_0\) :

The Reeb field

\(v_1,v_2\) :

Subriemannian orthonormal basis

\(\alpha _0,\alpha _1,\alpha _2\) :

Dual basis of \(v_0,v_1,v_2\)

\(\theta \) :

Tautological 1-form on \(T^*M\)

\(\omega \) :

Standard symplectic 2-form on \(T^*M\)

\(X\) :

A tangent vector in \(TT^*M\)

\(\mathcal L\) :

Lie derivative

\(\nabla _H\) :

Horizontal gradient

\(\Delta _H\) :

Sub-Laplacian

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Acknowledgments

We thank Igor Zelenko and Cheng-Bo Li for very interesting and stimulating discussions. This work is part of the PhD thesis of the second author. He would like to express deep gratitude to his supervisor, Boris Khesin, for his continuous support. He would like to thank Professor Karl-Theodor Sturm for the fruitful discussions. He is also grateful to SISSA for their kind hospitality where part of this work is done. Finally, we would also like to thank the referees for providing many constructive comments.

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Authors and Affiliations

  1. International School for Advanced Studies, via Beirut 4, 34014 , Trieste, Italy

    Andrei Agrachev

  2. Steklov Mathematical Institute, ul. Gubkina 8, Moscow, 119991, Russia

    Andrei Agrachev

  3. The Chinese University of Hong Kong, Room 216, Lady Shaw Building, Shatin, Hong Kong

    Paul W. Y. Lee

Authors
  1. Andrei Agrachev
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  2. Paul W. Y. Lee
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Correspondence to Paul W. Y. Lee.

Additional information

The first author was partially supported by PRIN and the second author was supported by NSERC postgraduate scholarship and postdoctoral fellowship.

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Agrachev, A., Lee, P.W.Y. Generalized Ricci curvature bounds for three dimensional contact subriemannian manifolds. Math. Ann. 360, 209–253 (2014). https://doi.org/10.1007/s00208-014-1034-6

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  • DOI: https://doi.org/10.1007/s00208-014-1034-6

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