PTR-submanifolds of a Kähler Manifold

Abstract In this paper, we introduce a new submanifold of a Kähler manifold. This new class of submanifolds includes totally real submanifolds, proper CR submanifolds, hemi-slant submanifolds and pointwise hemi-slant submanifolds as special classes. After giving examples for this new class of submanifolds, the geometry of distributions and their maximal integral manifolds is investigated. Then, the effect of morphisms defined by distributions on the geometry of the submanifold is studied. In the case that the ambient space in which the submanifold is contained is a complex space form, we determine special cases where both the submanifold and the ambient space are reduced. Furthermore, we determine the position of the mean curvature vector field when the submanifold is totally umbilical, and with the help of this, the geometric structure of the submanifold is obtained. Finally, if this new class of submanifolds is a Riemannian product manifold, the non-existence of this product submanifold in complex hyperbolic space is shown and an inequality related to the second fundamental form for complex projective space is given. It is shown that the factors of the Riemannian product manifold are also totally geodesic in complex projective space if the inequality becomes the equality.