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Rigidité des hypersurfaces en géométrie riemannienne et spinorielle : Aspect extrinsèque et intrinsèque

Abstract : In this thesis, we study the relation between extrinsic and intrinsic aspects for hypersurfaces of space forms by the way of rigidity results. First, we prove some pinching results for lower bounds of the extrinsic radius in terms of the r-th mean curvatures. We show that if the equality is almost achieved for one of these inequalities, then the hypersurface is close to a sphere (for the Hausdorff distance or diffeomorphic and almost-isometric). Then, we prove such results for upper bounds of the first eigenvalue of the Laplacian in Euclidean space, which give us some results about almost Einstein hypersurfaces. In a second time, we give a spinorial charcterization of surfaces into 3-homogenous manifolds with 4-dimensional isometry group. Namely, we show that the existence of a special spinor field on the surface is a necessary and sufficient condition to be isometrically immersed in such a 3-homogeneous manifold.
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  • HAL Id : tel-01748157, version 1

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Julien Roth. Rigidité des hypersurfaces en géométrie riemannienne et spinorielle : Aspect extrinsèque et intrinsèque. Mathématiques générales [math.GM]. Université Henri Poincaré - Nancy 1, 2006. Français. ⟨NNT : 2006NAN10161⟩. ⟨tel-01748157v1⟩

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