Normal Subtwistor Structures

This paper introduces the concept of normal subtwistor structure. It is proved that any manifold that admits a normal subtwistor structure is locally isometric to direct product of Hermitian submanifold and Riemannian submanifold. However, it is locally isometric to the direct product of Kahler submanifold and Riemannian submanifold when the normal subtwistor structure has a closed fundamental 2-form. It is shown that a normal subtwistor structure on a real manifold of arbitrary dimension induces a sub-Kahler structure on this manifold, and all the integral submanifolds of the bundle are Kahler submanifolds. The author described in the previous works the case when there is a class of normal subtwistor structure examples on a Lie group. The concept of torsion tensor of a subtwistor structure is introduced, and it is shown that a normal subtwistor structure always has a vanishing torsion tensor. The obtained results help describe the local geometry of the manifold with normal subtwistor structure.