# Special submanifolds of Spin$^c$ manifolds

1 Géométrie
IECL - Institut Élie Cartan de Lorraine
Abstract : In this thesis, we aim to make use of Spin$^c$ geometry to study special submanifolds. We start by establishing basic results for the Spin$^c$ Dirac operator. We give then inequalities of Hijazi type involving the energy-momentum tensor. Studying the energy-momentum tensor on a Spin$^c$ manifold is related to several geometric situations. Indeed, it appears in the study of the variations of the spectrum of the Dirac operator and in the Einstein-Dirac equation. The study of hypersurfaces of Spin$^c$ manifolds allows us for a better understanding of this tensor since it is the second fundamental form of the immersion. Being natural structures on the 3-homogeneous manifolds $E(\kappa, \tau)$ , Spin$^c$ structures will be investigated in the study of some Riemannian problems on hypersurfaces of these manifolds. In fact, we prove a Lawson correspondence for constant mean curvature surfaces in $\Ekt$. Finally, we characterize complex structures and CR structures by $\Spinc$ structures admitting a special spinor, called pure spinor or transversal spinor.
Keywords :
Document type :
Theses
Mathematics [math]. Université Henri Poincaré, Nancy 1, 2011. English
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https://tel.archives-ouvertes.fr/tel-00624641
Contributor : Roger Nakad <>
Submitted on : Monday, January 2, 2012 - 3:59:21 PM
Last modification on : Tuesday, February 23, 2016 - 1:04:46 AM

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• HAL Id : tel-00624641, version 2

### Citation

Roger Nakad. Special submanifolds of Spin$^c$ manifolds. Mathematics [math]. Université Henri Poincaré, Nancy 1, 2011. English. <tel-00624641v2>

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