Spinorial characterizations of surfaces into three-dimensional homogeneous manifolds

Spinorial characterizations of Surfaces into 3-dimensional homogeneous Manifolds

We give a spinorial characterization of isometrically immersed surfaces into 3-dimensional homogeneous manifolds with 4-dimensional isometry group in terms of the existence of a particular spinor, called generalized Killing spinor. This generalizes results by T. Friedrich [7] for R3 and B. Morel [16] for S3 and H3. The main argument is the interpretation of the energy-momentum tensor of a genra...

Isometric Immersions into 3-dimensional Homogeneous Manifolds

We give a necessary and sufficient condition for a 2-dimensional Riemannian manifold to be locally isometrically immersed into a 3-dimensional homogeneous manifold with a 4-dimensional isometry group. The condition is expressed in terms of the metric, the second fundamental form, and data arising from an ambient Killing field. This class of 3-manifolds includes in particular the Berger spheres,...

Low-dimensional Homogeneous Einstein Manifolds

A closed Riemannian manifold (Mn, g) is called Einstein if the Ricci tensor of g is a multiple of itself; that is, ric(g) = λ · g. This equation, called the Einstein equation, is a complicated system of second order partial differential equations, and at the present time no general existence results for Einstein metrics are known. However, there are results for many interesting classes of Einst...

Higher Order Parallel Surfaces in Three-dimensional Homogeneous Spaces

We give a full classification of higher order parallel surfaces in three-dimensional homogeneous spaces with four-dimensional isometry group, i.e. in the so-called Bianchi-CartanVranceanu family. This gives a positive answer to a conjecture formulated in [2]. As a partial result, we prove that totally umbilical surfaces only exist if the space is a Riemannian product of a surface of constant Ga...

Locally Tame Curves and Surfaces in Three-dimensional Manifolds