Fundamental theorem for totally complex submanifolds (Q2469782)

\textit{D. V. Alekseevsky} and \textit{S. Marchiafava} [Osaka J. Math. 38, No. 4, 869--904 (2001; Zbl 0998.53032)] conjectured that the fundamental theorem of submanifolds geometry holds for half-dimensional totally complex submanifolds of both the quaternionic projective space and the quaternionic hyperbolic space. The author states a fundamental theorem for some families of submanifolds in the framework of Grassmann geometries, in a unified way. As an application he obtains a fundamental theorem for half-dimensional totally real submanifolds of a complex projective space and of a complex hyperbolic space. Finally, he shows that the Alekseevsky-Marchiafava conjecture is in the affirmative.