Ehresmann connections for a foliated manifold and certain kinds of rectangles without terminal vertex

Abstract

We define the notion of a non-extendable rectangle without terminal vertex for a foliated manifold (M, \mathfrak{F}) with a complementary distribution D and classify them into non-singular ones and singular ones. It is easy to show that D is an Ehresmann connection in the sense of R. A. Blumenthal and J. J. Hebda if and only if there is no non-extendable rectangle without terminal vertex. One of our purposes is to investigate the existence of singular non-extendable rectangle without terminal vertex. Another purpose is to obtain a new sufficient condition for the orthogonal complementary distribution of a foliation on a Riemanman manifold to be an Ehresmann connection by investigating a property of singular non-extendable rectangles without terminal vertex.