By Philippe Tondeur (auth.)
The subject matters during this survey quantity trouble examine performed at the differential geom etry of foliations over the past few years. After a dialogue of the fundamental ideas within the thought of foliations within the first 4 chapters, the topic is narrowed all the way down to Riemannian foliations on closed manifolds starting with bankruptcy five. Following the dialogue of the particular case of flows in bankruptcy 6, Chapters 7 and eight are de voted to Hodge conception for the transversal Laplacian and functions of the warmth equation way to Riemannian foliations. bankruptcy nine on Lie foliations is a prepa ration for the assertion of Molino's constitution Theorem for Riemannian foliations in bankruptcy 10. a few features of the spectral conception for Riemannian foliations are mentioned in bankruptcy eleven. Connes' viewpoint of foliations as examples of non commutative areas is in brief defined in bankruptcy 12. bankruptcy thirteen applies rules of Riemannian foliation idea to an infinite-dimensional context. other than the record of references on Riemannian foliations (items in this checklist are said within the textual content by way of [ ]), we now have integrated a number of appendices as follows. Appendix A is a listing of books and surveys on specific elements of foliations. Appendix B is an inventory of complaints of meetings and symposia committed partly or fullyyt to foliations. Appendix C is a bibliography on foliations, which makes an attempt to be a fairly whole record of papers and preprints almost about foliations as much as 1995, and comprises nearly 2500 titles.
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Additional resources for Geometry of Foliations
3 is the case corresponding to q = 1. For a holonomy invariant l/ we have dl/ = 0 and 0: = 0 is a suitable choice, hence the De Rham class in question vanishes. If however Ex r F ---t B arises from a representation of r by isometries of a Riemannian metric gF on F, then this will turn F into a Riemannian foliation. Consider again an arbitrary foliation on (M, g), with induced metric gQ on Q. 18) (8(Y)gQ)(X,X') = YgQ(X, X') - gQ(7f[Y, X], X') - gQ(X,7f[Y,X']), where X, X' E r LJ... Therefore (8(Y)gQ)(X,X') = Yg(X,X') - g([Y,X],X' ) - g(X, [Y,X']) = g(V'WY,X ' ) + g(X, V'W,Y).
FOT Y E V(F) and X, X' E 48 5 TRANSVERSAL RIEMANNIAN GEOMETRY An infinitesimal automorphism Y is transversally metric, if 8(Y)gQ = o. If this holds, Y = 1f(Y) is called a transversal Killing field (Molino [Mo7,8]). For the point foliation with L = 0 this is the usual definition of a Killing vector field. 22). Thus ~8(Y)gQ is the symmetric part of the 2-linear form V'w on Q. A transversally oriented Riemannian foliation has a canonical holonomy invariant transversal volume v. We state the following fact.
15 PROPOSITION. a is symmetric. Proof. a(X, Y) - a(Y, X) = -\7 X7r(Y) + \7Y7r(X) +7r[X, Y] = -T\7(X, Y), which vanishes. D In the sequel we will use the symbol a for the restriction of this form to L. 16) for U, V E fL. a is the second fundamental form of the leaves of F in (M, g). The equation a = 0 (along L) holds if and only if each leaf of F is a totally geodesic submanifold of (M,g). To give an interpretation of the second fundamental form, consider Y E f Ll.. which is an infinitesimal automorphism of F.
Geometry of Foliations by Philippe Tondeur (auth.)