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Rigidity Results for Riemannian Spin\(^c\) Manifolds with Foliated Boundary

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Abstract

Given a Riemannian spin\(^c\) manifold whose boundary is endowed with a Riemannian flow, we show that any solution of the basic Dirac equation satisfies an integral inequality depending on geometric quantities, such as the mean curvature and the O’Neill tensor. We then characterize the equality case of the inequality when the ambient manifold is a domain of a Kähler–Einstein manifold or a Riemannian product of a Kähler–Einstein manifold with \(\mathbb R\) (or with the circle \(\mathbb S^1\)).

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Acknowledgements

Part of this work was done while Fida El Chami and Georges Habib enjoyed the hospitality of the University of Lorraine. The second-named author would like to thank the Alexander von Humboldt Foundation for its support. We also want to thank the referee for many remarks that improved the presentation of the paper.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Sciences II, Lebanese University, P.O. Box 90656, Fanar-Matn, Lebanon

    Fida El Chami & Georges Habib

  2. Institut Élie Cartan de Lorraine, Université de Lorraine, Site de Metz, rue Augustin Fresnel, 57070, Metz, France

    Nicolas Ginoux

  3. Department of Mathematics and Statistics, Faculty of Natural and Applied Sciences, Notre Dame University-Louaizé, P.O. Box 72, Zouk Mikael, Lebanon

    Roger Nakad

Authors
  1. Fida El Chami
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  2. Nicolas Ginoux
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  3. Georges Habib
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  4. Roger Nakad
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Correspondence to Roger Nakad .

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El Chami, F., Ginoux, N., Habib, G. et al. Rigidity Results for Riemannian Spin\(^c\) Manifolds with Foliated Boundary. Results Math 72, 1773–1806 (2017). https://doi.org/10.1007/s00025-017-0734-0

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  • DOI: https://doi.org/10.1007/s00025-017-0734-0

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