# Hokkaido Mathematical Journal

## Hokkaido Mathematical Journal, 36 (2007) pp.641-668

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### Abstract

The fundamental theorem (existence and uniqueness) for submanifolds of real space forms is well-known. We will discuss 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 projective space $\mathbb{H}P^n$ or the quaternion hyperbolic space $\mathbb{H}H^n$. This result is an affirmative answer to the conjecture by Alekseevsky and Marchiafava.

MSC(Primary) 53B25 53C26 the fundamental theorem for submanifolds, totally complex submanifolds