Totally geodesic submanifolds in Lie groups
Let \(F\) be the set of real numbers \(\mathbb{R}\), complex numbers \(\mathbb{C}\) or quaternions \(\mathbb{H}\), and for, any integer \(N\), denote by \(G_F(N)\) the Lie group acting on the left of \(F^{N}\) and preserving the usual inner product of \(F^N\). So \(G_\mathbb{R}(N)\) is the orthogonal group \(O(N)\), \(G_\mathbb{C}(N)\) is the complex unitary group \(U(N)\), and \(G_\mathbb{H}(N)\) is the symplectic group Sp\((N)\). In the nice paper under review, the author proves there is a natural imbedding of the Grassmann manifold \(G_F(n,N)\) in the Lie group \(G_F(N)\) which makes \(G_F(n,N)\) a totally geodesic submanifold, and study some properties of this imbedding. Similarly one shows there is a natural imbedding \(S^7\to \text{SO}(8)\) defined by octonians which makes \(S^7\) a totally geodesic submanifold in SO(8). Next the author proves that the natural inclusion of \(G_F(N)\) in the sphere \(S^{cN^2-1}(\sqrt{ N})\subset \text{gl}(N,F)\) (\(c=\) real dimension of \(F\)) is minimal, and therefore the natural imbedding \(G_F(N)\to \text{gl}(N,F)\) is formed by the eigenfunctions of the Laplacian on \(G_F(N)\) [\textit{T. Takahashi}, J. Math. Soc. Japan, 18, 380--385 (1966; Zbl 0145.18601)] one knows that a compact homogeneous Riemannian manifold with irreducible linear isotropy group admits a minimal immersion into a Euclidean sphere). One may prove similar results for the special unitary group \(\text{SU}(N)\) and the exceptional Lie groups \(\text{Spin}_7\) and \(G_2\).
scientific article; zbMATH DE number 5082257