Normal homogeneous spaces admitting totally geodesic hypersurfaces (Q1127016)

A normal homogeneous space is a reductive homogeneous manifold \(G/K\) endowed with a \(G\)-invariant Riemannian metric induced by a biinvariant inner product on the compact Lie group \(G\) (e.g., the opposite of the Killing-Cartan form). In this paper, the author proves that if a normal homogeneous space \(G/K\), with \(G\) compact simple, admits a totally geodesic hypersurface, then \(G/K\) has (positive) constant sectional curvature. This generalizes a result of \textit{B.-Y. Chen} and \textit{T. Nagano} [Duke Math. J. 45, 405-425 (1978; Zbl 0384.53024)], asserting that the sphere is the only compact symmetric space admitting a totally geodesic hypersurface. Moreover, if \(G/K\) is a homogeneous space with \(G\) and \(K\) of the same rank (which is equivalent to \(G/K\) having positive Euler characteristic), the author proves that \(G/K\) has (positive) constant sectional curvature under the weaker assumption that the tangent space to \(G/K\) admits a curvature invariant hyperplane. A crucial role in the proof is played by the description of the Lie algebra of \(K\) in terms of the root system of the Lie algebra of \(G\).