Hipersuperfícies de espaços homogêneos e de grupos de Lie lorentzianos e deformações de métricas kählerianas.

Abstract

This work is divided into two parts. In the first part, we obtain a rigidity theorem for complete CMC hypersurfaces in a Riemannian homogeneous space, with versions for compact and noncompact hypersurfaces. In the case of a Lorentzian Lie group G with a bi-invariant metric, we show that the only compact CMC spacelike hypersurfaces immersed in G are the lateral classes of a Lie subgroup L of G. We also show that every compact spacelike hypersurfaces of G having positive semi-definite Weingarten operator A are the totally geodesic ones. In the second part, from a Kählerian manifold M with a closed conformal vector field, we crafted a family of Kählerian metrics of M and get ordinary differential equations to find examples of Einstein metrics and Ricci solitons in the Euclidean complex space and in a Riemannian cone over a Sasaki Einstein manifold.