Geometry of Nearly Sasakian Manifolds and their Submanifo

Abstract

This thesis is divided into two different parts. The first is about the study of the surface in the sphere of dimension five, with the nearly Sasakian structure or the cosymplectic structure. The second part is the classification of 4-dimensional locally strongly convex homogeneous affine hypersurfaces, in the case how the shape operator have two distinct eigenvalues, by considering that the multiplicity of both eigenvalues is 2. Part I : We investigate surfaces in the nearly Sasakian 5-sphere for which the structure vector field ξ is normal to the surface and which are anti-invariant with respect to the nearly Sasakian structure. We show the flowing theorem : Theorem : A totally real surface of the nearly Sasakian S 5 is always minimal. We show also that this result is also valid for the surfaces in nearly cosymplectic 5-sphere. As a consequence of the minimality, we can also obtain a local correspondence between totally real surfaces of the S 5 with nearly Sasakian structure, or nearly cosymplectic structure, and minimal Lagrangian surfaces of the complex projective space CP 2 . Part II : We study 4 dimensional locally strongly convex, locally homogeneous, hypersur faces whose affine shape operator has two distinct principal curvatures. In case that one of the eigenvalues has dimension 1 these hypersurfaces have been previously studied by Dillem, Vrancken, Hu, Li and Zhang, in which a classification of such submanifolds was obtained in dimension 4 and 5 under the additional assumption that the multiplicity of one of the eigenvalues is 1. We complete the classification in dimension 4 by considering the case that the multiplicity of both eigenvalues is 2, this is the theorem : Theorem : Let M4 be a locally strongly convex, locally homogeneous, affine hy persurface in ❘5 . Assume that M has two distinct eigenvalues, both of multiplicity 2. Then M is equivalent to the convex part of one of the following hypersurfaces: