Sugli spazi omogenei di dimensione tre SO(2) - isotropi

In this thesis we studied some problems from the theory of the submanifolds of the three-dimensional Riemannian manifolds. Our intention is to evaluate which properties of the submanifolds depend by the dimension of the group of isometries. We considered a two-parameter family of three-dimensional Riemannian manifolds (M, ds2 ?,m), endowed with the Cartan - Vranceanu metrics. These metrics can be found in the classification of 3-dimensional homogeneous metrics given by L. Bianchi. Their geometric interest lies in the following fact: the family of metrics includes all 3-dimensional homogeneous metrics whose group of isometries has dimension 4 or 6, except for those of constant negative sectional curvature. The group of isometries of these spaces has a subgroup isomorphic to the group SO(2), so there exist surfaces of revolution around z-axis. We explicitly obtained the Lie algebra of the Killing vector fields and thus the group of isometries for the C-V metrics. We determined the equations of the geodesics using the Killing vector fields and obtain explicitly the equation of the surface which con- tains the geodesics. After having determined the totally geodesics surfaces isometrically immersed in the C-V spaces, we studied the totally umbili- cal submanifolds of these spaces, proving that the only totally umbilical submanifolds are totally geodesic. We found the geodesics for the SO(2)- invariant surfaces of the Cartan-Vranceanu spaces, deduced the conditions that meridians and parallels must satisfy in order to be geodesics and show the analogies with the euclidian case.