Download PDF by Philippe Tondeur (auth.): Geometry of Foliations

By Philippe Tondeur (auth.)

ISBN-10: 3034889143

ISBN-13: 9783034889148

ISBN-10: 3034898258

ISBN-13: 9783034898256

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

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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.

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Geometry of Foliations by Philippe Tondeur (auth.)

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Categories: Algebraic Geometry