Totally geodesic submanifolds in Lie groups

Summary

We study minimal and totally geodesic submanifolds in Lie groups and related problems. We show that: (1) The imbedding of the Grassmann manifold G _F (n,N) in the Lie group G _F (N) defined naturally makes G _F (n,N) a totally geodesic submanifold; (2) The imbedding S ⁷→SO(8) defined by octonians makes S ⁷a totally geodesic submanifold inSO(8); (3) The natural inclusion of the Lie group G _F (N) in the sphere S^{cN^2-1}(√N) of gl(N,F)is minimal. Therefore the natural imbedding G _F (N)<span style='font-size:10.0pt;font-family:"Lucida Sans Unicode"'>→gl(N,F )is formed by the eigenfunctions of the Laplacian on G _F (N).