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Curvature-dimension inequalities on sub-Riemannian manifolds obtained from Riemannian foliations: part I

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Abstract

We give a generalized curvature-dimension inequality connecting the geometry of sub-Riemannian manifolds with the properties of its sub-Laplacian. This inequality is valid on a large class of sub-Riemannian manifolds obtained from Riemannian foliations. We give a geometric interpretation of the invariants involved in the inequality. Using this inequality, we obtain a lower bound for the eigenvalues of the sub-Laplacian. This inequality also lays the foundation for proving several powerful results in Part II.

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Acknowledgments

This work has been supported by the Fonds National de la Recherche Luxembourg (AFR 4736116 and OPEN Project GEOMREV).

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  1. Mathematics Research Unit, FSTC, University of Luxembourg, 6, rue Richard Coudenhove-Kalergi, 1359, Luxembourg, Grand Duchy of Luxembourg

    Erlend Grong & Anton Thalmaier

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  1. Erlend Grong
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  2. Anton Thalmaier
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Correspondence to Erlend Grong.

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Grong, E., Thalmaier, A. Curvature-dimension inequalities on sub-Riemannian manifolds obtained from Riemannian foliations: part I. Math. Z. 282, 99–130 (2016). https://doi.org/10.1007/s00209-015-1534-4

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