  In this paper, we will first derive a DDVV-type optimal inequality for real
skew-symmetric matrices, then we apply it to establish a Simons-type integral
inequality for Riemannian submersions with totally geodesic fibres and
Yang-Mills horizontal distributions. In this way, we show phenomenons of
duality between Submanifold geometry and Riemannnian submersion, particularly
between second fundamental form of a submanifold and integrability tensor of a
Riemannian submersion.
