Ricci curvature bounds and rigidity for non-smooth Riemannian and semi-Riemannian metrics
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Bibliographic record
Abstract
Abstract We study rigidity problems for Riemannian and semi-Riemannian manifolds with metrics of low regularity. Specifically, we prove a version of the Cheeger-Gromoll splitting theorem [22] for Riemannian metrics and the flatness criterion for semi-Riemannian metrics of regularity $$C^1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>C</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:math> . With our proof of the splitting theorem, we are able to obtain an isometry of higher regularity than the Lipschitz regularity guaranteed by the $$\textsf{RCD}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>RCD</mml:mi> </mml:math> -splitting theorem [30, 31]. Along the way, we establish a Bochner-Weitzenböck identity which permits both the non-smoothness of the metric and of the vector fields, complementing a recent similar result in [62]. The last section of the article is dedicated to the discussion of various notions of Sobolev spaces in low regularity, as well as an alternative proof of the equivalence (see [62]) between distributional Ricci curvature bounds and $$\textsf{RCD}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>RCD</mml:mi> </mml:math> -type bounds, using in part the stability of the variable $$\textsf{CD}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>CD</mml:mi> </mml:math> -condition under suitable limits [47].
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