  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.
