WINTGEN INEQUALITIES ALONG RIEMANNIAN SUBMERSIONS

In this paper, a Wintgen inequality is obtained depending on O'Neill's tensor field Talong a Riemannian submersion from a real space form to a Riemannian manifold and the geometric meaning of the equality case is provided. Then, a Wintgen inequality is derived along a Riemannian submersion from a complex space form to Riemannian manifold, and a geometric result is provided in the case of equality. In addition, a Wintgen inequality is obtained using concepts based on O'Neill's tensor field A, and it is shown that the condition for equality is essentially equivalent to the integrability of the horizontal distribution. This condition is also investigated in the case of a complex space form.