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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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
Keywords
- Manifolds with boundary
- spin\(^c\) structures
- Riemannian flows
- basic Dirac equation
- Kähler–Einstein manifolds
- parallel spinors