New curvature tensors along Riemannian submersions

TL;DR

This paper introduces new curvature tensors along Riemannian submersions, extending classical curvature relations, and explores their properties when the total space has umbilical fibers.

Contribution

It defines several new curvature tensors along Riemannian submersions and investigates their behavior in the case of umbilical fibers, expanding the understanding of curvature relations.

Findings

Derived relations for new curvature tensors in Riemannian submersions.

Established results for total spaces with umbilical fibers.

Extended classical curvature equations to new tensor contexts.

Abstract

In 1966, B. O'Neill [The fundamental equations of a submersion, Michigan Math. J., Volume 13, Issue 4 (1966), 459-469.] obtained some fundamental equations and curvature relations between the total space, the base space and the fibres of a submersion. In the present paper, we define new curvature tensors along Riemannian submersions such as Weyl projective curvature tensor, concircular curvature tensor, conharmonic curvature tensor, conformal curvature tensor and $M-$projective curvature tensor, respectively. Finally, we obtain some results in case of the total space of Riemannian submersions has umbilical fibres for any curvature tensors mentioned by the above.