On totally umbilical QR-submanifolds of quaternion Kaehlerian manifolds

Aurel Bejancu, Hani Reda Farran
2000 Bulletin of the Australian Mathematical Society  
We introduce the notion of generalised 3-Sasakian structure on a manifold and show that a totally umbilical, but not totally geodesic, proper QR-submanifold of a quaternion Kaehlerian manifold is an extrinsic sphere and inherits such a structure. where v L is the complementary orthogonal vector bundle to v in TM L . If in particular, v = TM L or v -{0} we say that M becomes a quaternionic submanifold (see Chen [6]) or an anti-quaternionic submanifold (see Pak [10]). In the case M is a real
more » ... se M is a real hypersurface of M we have g{J a N,N) -0 for any a 6 {1,2,3} and normal vector field /V. Hence JaiTM-1 ) C TM, that is, M is an example of Qfi-submanifold with v -{0}. Suppose M is a Qfi-submanifold of M which is not a quaternionic submanifold. Then for each x € M, we denote J a {Vx) by A n , a. G {1,2,3}. It is easy to see that Di x ,D 2x , and D 3x are mutually orthogonal subspaces of T X M and have the same dimension s as available at https://www.cambridge.org/core/terms. https://doi.
doi:10.1017/s0004972700018517 fatcat:u2gr4bkzazd6lmaszimc27j5di