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

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• 作者：Erlend Grong ; Anton Thalmaier
• 关键词：Curvature ; dimension inequality ; Sub ; Riemannian geometry ; Hypoelliptic operator ; Spectral gap ; Riemannian foliations ; 58J35 ; 53C17 ; 58J99
• 刊名：Mathematische Zeitschrift
• 出版年：2016
• 出版时间：February 2016
• 年：2016
• 卷：282
• 期：1-2
• 页码：99-130
• 全文大小：685 KB
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• 作者单位：Erlend Grong (1)
  Anton Thalmaier (1)
  
  1. Mathematics Research Unit, FSTC, University of Luxembourg, 6, rue Richard Coudenhove-Kalergi, 1359, Luxembourg, Grand Duchy of Luxembourg
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-1823

文摘

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. Keywords Curvature-dimension inequality Sub-Riemannian geometry Hypoelliptic operator Spectral gap Riemannian foliations