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Fundamental theorem for totally complex submanifolds

The fundamental theorem (existance and uniqueness) for or submanifolds of real space forms is well-known. We will diseuse this theorem for some families of submanifolds in the framework of Grassmann geometries in a unified way. In particular, we show the fundamental theorem for half dimensional totally complex submanifolds of the quaternion projetive space ∃P n or the quaternion hyperbolic space EH n . This result is an affirmative answer to the conjocture by Alekseevsky and Marchiafava.