Imersões isométricas em variedades homogêneas de dimensão 3

Abstract

A classical problem in geometry is to find conditions for one a manifold to be immersed isometrically in another. In this work, we present necessary and sufficient conditions for a simply connected 2-dimensional Riemannian manifold to be immersed isometrically into a 3-dimensional homogeneous simply connected Riemannian manifold with a 4-dimensional isometry group. We will see that such conditions are expressed in terms of the metric, the second fundamental form, and data arising from an ambient Killing field. This result was obtained by Benoît Daniel in the paper entitled "Isometric immersions into 3-dimensional homogeneous manifolds" and has relevant resultads for the differential geometry. The tools to demonstrate this theorem are based on use of the thechnique of moving frame and integrable distributions.