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.

中文翻译：

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向量束上球对称度量的曲率

本文致力于研究配备球对称度量的矢量束流形的几何形状。我们将计算曲率张量，截面曲率，里氏曲率张量和标量曲率。然后，我们将描述恒定截面曲率的球对称度量。此外，我们将建立有关标量曲率和Ricci曲率的刚度结果。最后，我们使用潜水的基本张量研究了大地测量学。