On the geometry of the normal bundle with a metric of Cheeger-Gromoll type

We investigate the geometry of a normal bundle equipped with a $(p,q)$-metric, i.e., Riemannian metric of Cheeger-Gromoll type, to the submanifold of a Riemannian manifold. We derive all natural object as the Levi-Civita connection, curvature tensor, sectional and scalar curvature. We prove that under some natural conditions the sectional curvature of this bundle may be bounded from below by given arbitrary large positive constant. Next we investigate $(p,q)$-metrics from the complex geometry point of view. We show when the normal bundle can by equipped with a structure of almost Hermitian, almost K\"ahlerian, conformally almost K\"ahlerian or K\"ahlerian manifold.