Grassmann geometries on compact symmetric spaces of general type
Let \(M\) be an \(m\)-dimensional smooth manifold and \(s\) a positive integer such that \(1\leq s\leq m\). Let \(G^s(TM)\) be the Grassmann bundle over \(M\) with the Grassmann manifold of \(s\)-planes as each fibre. Given a subset \(\Sigma\) of \(G^s(TM)\), an \(s\)-dimensional connected submanifold \(S\) of \(M\) is called a \(\Sigma\)-submanifold if all the tangent spaces of \(S\) belong to \(\Sigma\). The collection of \(\Sigma\)-submanifolds constitutes a so-called \(\Sigma\)-geometry, a Grassmann geometry introduced by \textit{R. Harvey} and \textit{H. B. Lawson} [Acta Math. 148, 47-157 (1982; Zbl 0584.53021)]. In this paper we assume that \(M\) is a compact simply connected Riemannian symmetric space and \(\Sigma\) is a \(G\)-orbit satisfying the following condition, where \(G\) denotes the isometry group of \(M\) and acts canonically on \(G^s(TM)\): any element \(V\in\Sigma\) and its orthogonal complement \(V^{\bot}\) are preserved by the curvature tensor of \(M\). We consider such a \(\Sigma\)-geometry. A \(\Sigma\)-geometry always has a unique compact totally geodesic \(\Sigma\)-submanifold. The aim of this paper is to complete the classification of all \(\Sigma\)-geometries which admit non-totally geodesic \(\Sigma\)-submanifolds. As an application, we have the classification of symmetric submanifolds of \(M\).
scientific article; zbMATH DE number 1199730