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

TL;DR

This paper explores the geometric properties of normal bundles with Cheeger-Gromoll type metrics, deriving fundamental objects and analyzing curvature bounds, while also examining complex geometric structures on these bundles.

Contribution

It provides a comprehensive analysis of the Levi-Civita connection, curvature, and complex structures on normal bundles with $(p,q)$-metrics of Cheeger-Gromoll type.

Findings

Sectional curvature can be bounded from below by arbitrary large positive constants.

Conditions for the normal bundle to admit almost Hermitian, almost Kähler, or Kähler structures.

Derived explicit formulas for connection and curvature tensors in this setting.

Abstract

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.