Structural equations and an integral formula for foliated manifolds (Q1066474)

The following formula has been proved and applied: \[ \sum_{i,j}K(v_ i,x_ j)=div F-\sum_{j}tr(A^{v_ j})^ 2+div H- | T| \quad^ 2+| F|^ 2+| H|^ 2. \] Here, K is the sectional curvature of a Riemannian manifold M, \(\{v_ j\}\) and \(\{x_ i\}\) are orthogonal frames of an integrable subbundle \({\mathcal V}\subset TM\) and of its orthogonal complement \({\mathcal H}\), T and A are the second fundamental forms of \({\mathcal V}\) and \({\mathcal H}\), \(F=\sum A_{x_ i}x_ i\) and \(H=\sum T_{v_ j}v_ j.\) \{The reviewer [''An integral formula for a Riemannian manifold with two orthogonal complementary distributions'', Colloq. Math. (to appear)] obtained a very similar formula without assumptions of integrability of \({\mathcal V}.\}\)