Curvatures of Spherically Symmetric Metrics on Vector Bundles

This paper is devoted to study the geometry of vector bundle manifolds equipped with spherically symmetric metrics. We will compute the curvature tensor, the sectional curvature, the Ricci curvature tensor and the scalar curvature. We will then characterize spherically symmetric metrics of constant sectional curvature. Moreover, we will establish some rigidity results regarding the scalar curvature and the Ricci curvature. Finally, we investigate geodesics using the fundamental tensors of a submersion.