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Totally complex submanifolds of a quaternionic Kaehlerian manifold

We consider some kinds of submanifolds in a quaternionic Kaehlerian manifold M i. e., those invariant, totally real and totally complex submanifolds. These submanifolds are curvature invariant (see § 6). In M, invariant or totally real submanifolds are defined as quaternionic analogues of corresponding submanifolds in a Kaehlerian manifold ([6], [9]). It is known that each invariant submanifold of M is totally geodesic. But in general an n-dimensional totally real submanifold of M is not necessarily totally geodesic (see [3]). We shall define totally complex submanifold in § 2 and prove in § 2 that a totally complex submanifold is minimal. In the present paper, totally complex submanifolds of a quaternionic Kaehlerian manifold will be studied and the so-called pinching problem of the length of the second fundamental form will be discussed as quaternionic analogues of the arguments developed in [6] and [9] for submanifolds of a Kajehlerian manifold. Our main result is stated in the following main theorem which will be given in § 5. In the following theorem and hereafter, we mean by a complex space form (resp. a quaternionic space form) a Kaehlerian manifold with constant holomorphic sectional curvature (resp. a quaternionic Kaehlerian manifold with constant curvature).