Study on submanfolds of homogeneous spaces from the view of orbital Grassmann geometry

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This study is on the Grassmann geometry on the Riemannian homogeneous spaces. Our aim is to consider the classification problem of extrinsic homogeneous submanifolds of Riemannian symmetric spaces. For this, in this study, we examine the case where a Riemannian homogeneous space is a 3-dimensional unimodular Lie group with a left invariant metric. The 3-dimensional unimodular Lie groups are classified into six ones; the 3-dimensional vector group, the 3-dimensional Heisenberg group, the groups of rigid motions of the Eucliden 2-plane and the Minkowski 2-plane, the special unitary group SU (2), and the special linear group SL (2,R). Also, for each of them the geometric properties such as the curvatures, the isometry group, and m on, can be expressed concretely. In this study we in particular consider the Grassmann geometry on the spaces SU (2) and SL (2,R), while the cases of the Heisenberg group and the groups of rigid motions of the Eucliden 2-plane and the Minkowski 2-plane are clarify by H Naitoh, J. Inoguchi, and K Kuwabara. The obtained main results are the following.;for both the spaces SU (2) and SL (2,R),(1) the classification for all the orbits associated with Grassmann geometries on their spaces(2) the determination of the orbits whose Grassmann geometries are nonempty(3) the analysis on the surface theory for nonempty Grassmann geometries, in particular, (3-1) the settlement of the existence problem of totally geodesic surfaces, (3-2) the settlement of the existence problem of flat surfaces, (3-3) the settlement of the existence problem of minimal surface, (3-4) the settlement of the existence problem of constant mean curvature surfaces(4) the overview of Grassmann geometry on all the 3-dimensional unimodular Lie groups with left invariant metrics.These results will be appeared in forthcoming papers titled by "Grassmann geometry on the 3-dimensional unimodular Lie groups I, II".