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