By Philippe Tondeur (auth.)
The subject matters during this survey quantity challenge study performed at the differential geom etry of foliations over the past few years. After a dialogue of the elemental strategies within the conception 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 specific case of flows in bankruptcy 6, Chapters 7 and eight are de voted to Hodge concept for the transversal Laplacian and functions of the warmth equation solution 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 conception for Riemannian foliations are mentioned in bankruptcy eleven. Connes' perspective of foliations as examples of non commutative areas is in brief defined in bankruptcy 12. bankruptcy thirteen applies principles of Riemannian foliation concept to an infinite-dimensional context. other than the record of references on Riemannian foliations (items in this record are spoke of within the textual content by means of [ ]), we've got integrated a number of appendices as follows. Appendix A is an inventory of books and surveys on specific points of foliations. Appendix B is a listing of lawsuits of meetings and symposia dedicated partly or solely to foliations. Appendix C is a bibliography on foliations, which makes an attempt to be a fairly whole checklist of papers and preprints as regards to foliations as much as 1995, and comprises nearly 2500 titles.
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Extra resources for Geometry of Foliations
L on forms w, Wi E fY(M, Q). For any connection V' in Q and its associated exterior differentiation d\7 in n"(M, Q), there is a codifferential 15\7 : nr+1(M, Q) ~ fY(M, Q). It is given by 15\7W = (_1)nr+1 * d\7 * w. The evaluation formula for an orthonormal frame E 1 , ... ,En is as follows: n 15\7W = - L iEA V'EA W. A=l For this operator to be formally the adjoint of d\7, more compatibility conditions for V' and 9 are required. We leave this question aside for the moment, and evaluate the operator 15\7 on 7r E n1 (M, Q).
61) is well-defined as a vector field along ,. The same interpretation has to be given to several expressions in the calculation to follow. Since V is F-Jacobi along the geodesic " we have 7r~(V'rV'rV + RM (V, i'h) = 7r~(V'rV't1i' + V'rb, V]- V'~,"rli' - V'rV'ifi') = o.
G. 17) a flow of isometries. Associated to a : L 181 L ----t Q there is a shape operator or Weingarten map. 18) gQ(a(U, V),s) = g(W(s)U, V) for U, V E fL. Let 7r~ : TM ----t L denote the orthogonal projection corresponding to the decomposition TM = L ffi L~ (L~ ~ Q). 19) Thus the characteristic polynomial of W is a geometric invariant associated to:F. Of particular interest is TrW(s). It is linear in s, hence TrW E fQ*. 20) K(V) = 0 for V E f L, K(S) = Tr W(s) for s E fQ, where we have used the identification L~ ~ Q.
Geometry of Foliations by Philippe Tondeur (auth.)
Categories: Algebraic Geometry