By Philippe Tondeur (auth.)
A first approximation to the belief of a foliation is a dynamical procedure, and the ensuing decomposition of a website via its trajectories. this can be an concept that dates again to the start of the speculation of differential equations, i.e. the 17th century. in the direction of the top of the 19th century, Poincare built equipment for the research of worldwide, qualitative homes of recommendations of dynamical platforms in events the place specific answer equipment had failed: He stumbled on that the examine of the geometry of the distance of trajectories of a dynamical approach finds advanced phenomena. He emphasised the qualitative nature of those phenomena, thereby giving powerful impetus to topological equipment. A moment approximation is the belief of a foliation as a decomposition of a manifold into submanifolds, all being of an analogous size. right here the presence of singular submanifolds, equivalent to the singularities relating to a dynamical approach, is excluded. this is often the case we deal with during this textual content, however it is not at all a complete research. to the contrary, many events in mathematical physics most probably require singular foliations for a formal modeling. the worldwide examine of foliations within the spirit of Poincare used to be began basically within the 1940's, by means of Ehresmann and Reeb.
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Extra resources for Foliations on Riemannian Manifolds
Namely dw = a A w implies A w. = Xa(Y)·w + a(Y)·a(X)·w- _ i(Y)w = 0, it follows that °= da A w, hence locally da =, Thus ° i(X)w = => i(X)da = i(X),·w and thus i(Y)i(X)da = 0, if i(Y)w = 0. 7 that i(X)w = 0, i(Y)w = 0 => R(X,Y) = 0. In other words the curvature is zero along the leaves of the foliation defined by w. 22 In view of the observatio~s above, the Godbillon-Vey form a A da is therefore the exterior product of the connection form a and the corresponding curvature 2-form. This is in contrast to the Chern-Weil construction, where differential forms are constructed from the curvature form alone.
Let M be a smooth manifold, and J transversally orientable foliation of codimension q. 10). PROOF. During the proof we repeatedly use the following facts. 12) P= * where w1 , ... ,wq locally frame Q. 13) d, = L'j j=l A Wj . 5. 13) we find q da = L aj j=l A Wj , A dv = da A v. If a 31 and thus q (da)q+1 =(\ L j=l a. )q+1 J J contains in each term (q+1)-fold exterior products of forms wj , which all vanish since Aq+1Q* = O. To prove (ii), consider first two transversal orientations v,v'. v' Then = hv with a nowhere zero function h.
The situation for arbitrary q is clarified as follows. 9 PROPOSITION. codimension q. Let 1 be a transversally orientable foliation of Then there is a nonsingular global q-form v E rAqQ * such = W1A... AWq , where w1 , ... ,wq defined by v = O. that locally v is of the form v the tangent bundle L E TM is This is obvious in a distinguished chart. and The proof of the global statement follows from a partition of unity argument. Note that the transition function of the line bundle AqQ * is det gap' * transition function of Q.