Integral formulae on foliated symmetric spaces
Let \(M\) be a compact Riemannian manifold endowed with a foliation \({\mathcal F}\) of codimension 1 and a unit vector field \(N\) normal to \({\mathcal F}\). The paper under review provides an infinite family of formulas relating integrals of elementary symmetric functions of principal curvatures of the leaves to similar integrals of some algebraic invariants obtained from the Weingarten operator of the foliation and the curvature operator in the direction of \(N\). Formulas of this type are eventually obtained even for some foliations with singularities, in particular for singular Riemannian foliations. Earlier results can be recovered as special cases, for instance those by \textit{F. Brito}, \textit{R. Langevin} and \textit{H. Rosenberg} [J. Differ. Geom. 16, 19--50 (1981; Zbl 0472.53049)], \textit{A. Ranjan} [Geom. Dedicata 20, 85--91 (1986; Zbl 0578.53025)], and \textit{P. G. Walczak} [Colloq. Math. 58, No. 2, 243--252 (1990; Zbl 0766.53024)]. For the sake of simplicity, the main calculations in the present paper are carried out for locally symmetric spaces.
scientific article; zbMATH DE number 6005030