Pointwise almost h-semi-slant submanifolds

We introduce the notions of pointwise almost h-slant submanifolds and pointwise almost h-semi-slant submanifolds as a generalization of slant submanifolds, pointwise slant submanifolds, semi-slant submanifolds, and pointwise semi-slant submanifolds. We obtain a characterization and investigate the following: the integrability of distributions, the conditions for such distributions to be totally geodesic foliations, the properties of h-slant functions and h-semi-slant functions, the properties of nontrivial warped product proper pointwise h-semi-slant submanifolds. We also obtain the topological properties of proper pointwise almost h-slant submanifolds and give an inequality for the squared norm of the second fundamental form in terms of a warping function and a h-semi-slant function for a warped product submanifold of a hyperkahler manifold. Finally, we give some examples of such submanifolds.