An invariant formula for orthogonal distributions
We give an integral formula relating certain curvatures of a foliation to the Ricci curvature of the ambient space. In particular, if M is a space form, we have the integral formula: \[ (1/vol(M))\int _{M}(2K_ 2(x)+S(x))dM=(m-1)c, \] where \(\dim (M)=m\), \(K_ 2\) is the second curvature polynomial of the foliation (it is assumed that the orthogonal distribution to the foliation is integrable), S represents the intrinsic Ricci curvature averaged over the tangent space of the foliation, and c represents the sectional curvature of the space form. In addition, we show that if \(S^ 2\times S^ 2\) has a foliation with minimal leaves, then it is either the trivial cross product foliation or the orthogonal distribution is nowhere integrable. Similar results are included.
scientific article; zbMATH DE number 4026270
scientific article