Title: Homogeneous hypersurfaces in low dimensional homogeneous spaces
Abstract: Homogeneous spaces are defined as manifolds on which a group of isometries acts transitively; that is, given any two points of the manifold, there exists an isometry mapping one to the other. Accordingly, hypersurfaces (or surfaces in the three-dimensional case) are said to be extrinsically homogeneous when they arise as orbits of the action of a subgroup of the isometry group of the ambient space. For the class of all left-invariant metrics on a Lie group G, one can determine the maximal isometry group and, consequently, establish a general classification of homogeneous submanifolds. For a Lie group G endowed with a left-invariant metric, its isometry group can, in general, be described in terms of the semidirect product of G with the group of metric-preserving automorphisms. Thus, when the isometry group has the same dimension as G, the classification of homogeneous submanifolds can be done through the classification of subgroups of G, up to orbit equivalence.
The classification of codimension-one subgroups of G can, in turn, be obtained from the classification of codimension-one subalgebras of its Lie algebra. Since this approach initially yields only the homogeneous hypersurfaces passing through the identity of G, the construction of a foliation by equidistant hypersurfaces yields a more general classification of homogeneous hypersurfaces, up to isometries of the ambient space.
In this presentation, we will address the classification of homogeneous hypersurfaces in four-dimensional homogeneous spaces and of homogeneous surfaces in three-dimensional homogeneous spaces.
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