Download e-book for iPad: Geometry of Foliations by Philippe Tondeur (auth.)

Algebraic Geometry

By Philippe Tondeur (auth.)

The themes during this survey quantity quandary study performed at the differential geom­ etry of foliations during the last few years. After a dialogue of the fundamental innovations within the idea of foliations within the first 4 chapters, the topic is narrowed right down to Riemannian foliations on closed manifolds starting with bankruptcy five. Following the dialogue of the targeted 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 points of the spectral idea for Riemannian foliations are mentioned in bankruptcy eleven. Connes' standpoint of foliations as examples of non­ commutative areas is in short defined in bankruptcy 12. bankruptcy thirteen applies rules of Riemannian foliation thought to an infinite-dimensional context. apart from the record of references on Riemannian foliations (items in this checklist are noted within the textual content via [ ]), we have now incorporated a number of appendices as follows. Appendix A is an inventory of books and surveys on specific elements of foliations. Appendix B is an inventory of lawsuits of meetings and symposia dedicated partly or totally to foliations. Appendix C is a bibliography on foliations, which makes an attempt to be a pretty entire record of papers and preprints near to foliations as much as 1995, and comprises nearly 2500 titles.

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Then P (8(Y)xF)(El"'" Ep) = YXF(E 1 , ... , Ep) - L XF(E 1 , ... , 8(Y)Ei"'" Ep) i=1 = - Lg(8(Y)Ei ,Ei )XF(E 1 , ... ,Ep) = - L g([Y, E i ], Ei). On the other hand, p p TrLA y = Lg(AyEi,E;) = - Lg(\l~y,Ei) i=1 i=1 p p i=1 i=1 = - Lg(\l~Ei + [Ei,Y],Ei) = - Lg([Ei,Y],Ei ), where we have used g(\l~ E i , E i ) = ~ Y g(Ei, E i ) = O. Comparison proves the desired identity. D We consider two special cases. 20 COROLLARY. 18, and let Y E fL1.. 21) where Ii is the mean curvature one-form of:F (RummIer [Ru 1]).

It follows that H~(F) -I- o. 0 Next we calculate the value of the integral of divE Y for Y E V (F) in case of a Riemannian foliation. 24 THEOREM. Let F be a transversally oriented Riemannian foliation on a closed oriented Riemannian manifold (M,g). Let Y E V(F). Then (the global scalar product of the sections T and Y of Q). 26). Since 'Po E F 2 n p+ 1, and i(Y)v E Fq-1nq-1, it follows that the term i(Y)v /\ 'Po is of filtration degree q + 1, and hence vanishes. Thus diVE YJ1 = d(i(Y)v /\ XF) + '" /\ i(Y)v /\ XF· 5 Since K 1\ v TRANSVERSAL RIEMANNIAN GEOMETRY 49 ErA q+ 1 Q*, and hence vanishes, we have further 0= i(Y)KV - K 1\ i(Y)v.

The transverse orient defined invariant, by the orientability of Q. 2) holonomy invariant transversal metric gQ there is a corresponding transversal 8(V)v = 0 for all V E rL, volume form v E rAqQ* c ~q(M). e. 2) 8(V)v = 0 for all V E rL, q L vh,···, 8(V)so;,"" Sq) (8(V)V)(Sl"'" vh,···,and 8(V)so;,"" for Sl, ... ,Sq rQ. ForSq) q ==1VV(Sl,"" the conceptSq)of-a L transversal holonomySq) invariant where (8(V)V)(Sl"'" Sq) = VV(Sl,"" Sq) - 0;=1 q E 0;=1 invariant transversal metric. volume coincides with the concept of a holonomy simple a Riemannian is given by nonsingular Killing for Sl,A...

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