By Michael Spivak

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Extra info for A Comprehensive Introduction to Differential Geometry, Vol. 5, Third Edition

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8 Proposition. ). b) Transversally oriented foliations with a holonomy invariant transversal volume form X1. are characterized by the condition H 1. = 0. 2. 1) may be introduced in an analytical way. 2. 9 Definition. 20) is called a Riemannian almost-product structure. The eigenspaces of P for the eigenvalues +1 and -1 are called the vertical distribution Tl and the horizontal distribution Tz ; they are complementary orthogonal: T M = Tl EB T2. 10 Remark. 21) are orthoprojectors from T M onto Tl and Tz.

Relative nullity foliations on infinite dimensional submanifolds of a Hilbert space are investigated in [Okr1]. Introducing the notion of L -parallel Jacobi fields, P. Dombrowski developed a method to study the question of the completeness of totally geodesic foliations induced by the kernel of differential forms. Continuing these examinations, H. Reckziegel with S. Hiepko [Rec 2], [HR], and R. Boning [Bo 1] dealt with variations of circles in Riemannian manifolds to achieve results on the completeness of spherical foliations induced by the kernel of certain differential forms.

In some sense HD(L, x) is a globalization of the germinal holonomy group r (L , x) and naturally surjects onto it. 32 Definition. [iu 4] Let {L} be a foliation on a manifold M with an Ehresmann connection D. The D-graph (D-holonomy groupoid) r D (L ) is the collection of all triples (x, y, [a]), where x, y lie in the same leaf L, a is a (piecewise) smooth path from x to y in Land [a] is the equivalence class of a [two paths a, (3 with the same endpoint x, yare equivalent if a{3-1 E KD(L, x)]. 33 Remark.

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