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 have characterizations and investigate the integrability of distributions, the conditions for such distributions to be totally geodesic foliations, the mean curvature vector fields on totally umbilic submanifolds, the properties of h-slant functions and h-semi-slant functions, the properties of non-trivial warped product proper pointwise h-semi-slant submanifolds. We also obtain topological properties on proper pointwise almost h-slant submanifolds and give an inequality for the squared norm of the second fundamental form in terms of the warping function and h-semi-slant functions for a warped product submanifold in a hyperk\"ahler manifold. Finally, we give some examples of such submanifolds.