On certain classes of almost product structures.

Bochner's technique in the theory of Riemannian almost product structures

“…In different terminology an umbilical distribution [5] is called a nonholonomic spher e [6] since the second fundamental form of such a distribution is proportional to its metric form. One has THEOREM 3.…”

Conformal curvature for the normal bundle of a conformal foliation

“…-A conformal foliation is locally conformally riemannian, that is, we can assume that X = 0 in a neighborhood of m [4]. Then, one can choose local coordinates such that V is the vertical foliation Since the imposed conditions are obviously necessary, the theorem is proved.…”

“…We recall that a conformal foliation can be characterized in the following manner [4]. Let V be a foliation on M. It is conformal iff there exist some riemannian metric g on M (it defines 96 = V 1 ) and vertical 1-form X such that (L^g) (X, X) = g(X, X) X(A).…”

“…That is the Bott connection except the term -X(Q)X. Some properties derived in [4] for (3) are written in the language of double forms as follows:…”

Complex Product Structures and Affine Foliations

“…Later, almost product structures were considered in the context of pseudo-Riemannian geometry; namely, a pseudo-Riemannian almost product manifold is given by an almost product structure E and a pseudo-Riemannian metric g compatible with E in the sense that they satisfy g(E X, EY ) = g(X, Y ) for all vector fields X, Y (see [16]). Using the method developed by Gray and Hervella in [17], Naveira gave in [31] a classification of Riemannian almost product manifolds (i.e., g is definite positive) in terms of the tensor field ∇ g E, where ∇ g is the Levi-Civita connection associated to g. The work of Naveira was completed subsequently in [18,29,30], and some special classes were discussed in [5]. Almost product manifolds have found applications also in physics, see for instance [7,9,12].…”