×

zbMATH — the first resource for mathematics

Opérateurs transversalement elliptiques sur un feuilletage riemannien et applications. (Transversally elliptic operators for Riemannian foliations and appplications). (French) Zbl 0697.57014
The author develops the theory of transversely elliptic operators for a Riemannian foliation \({\mathcal F}\) of a manifold \({\mathcal M}\). Basic differential operators act on the presheaves of basic sections of so- called \({\mathcal F}\)-vector bundles. A basic differential operator D is transversely elliptic if its symbol \(\sigma(D)(x,\xi)\) is an isomorphism for all x of \({\mathcal M}\) and \(\xi\neq \emptyset\). Transversely elliptic operators appear to be Fredholm and admit a Hodge decomposition. They have some cohomology properties which allow, among others, to prove the following, very interesting result of Calabi-Yao type: If \({\mathcal F}\) is transversely Kählerian and a class c in the basic cohomology \(H^ 2({\mathcal M}/{\mathcal F})\) of \({\mathcal F}\) contains at least one basic Kähler form \(\gamma\) for which \((1/2\pi)\gamma\) represents the transverse Chern class of \({\mathcal F}\), then c contains a basic Kähler form \(\omega\) for which \(\gamma\) is the Ricci form of the transverse Kähler metric corresponding to \(\omega\).
Reviewer: P.Walczak

MSC:
57R30 Foliations in differential topology; geometric theory
58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
53C12 Foliations (differential geometric aspects)
PDF BibTeX XML Cite
Full Text: Numdam EuDML
References:
[1] M.F. Atiyah , Elliptic operators and compact groups . Lecture Notes in Math, no. 401 (1974). · Zbl 0297.58009 · doi:10.1007/BFb0057821
[2] E. Calabi , On Kähler manifolds with vanishing canonical class. Algebraic Geometry and Topology, A symposium in honor of Lefschetz , Princeton University Press (1955), 78-79. · Zbl 0080.15002
[3] Y. Carriere , Flots riemanniens . Journées sur les structures transverses des feuilletages, Toulouse, Astérisque no 116 (1984). · Zbl 0548.58033
[4] A. Connes , A survey of foliations and operator algebras . Proceedings of Symp. in Pure Math. Vol. 38 (1982). · Zbl 0531.57023
[5] A. El Kacimi-Alaoui et G. Hector , Décomposition de Hodge basique pour un feuilletage riemannien . Ann. Inst. Fourier de Grenoble 36, 3 (1986), 207-227. · Zbl 0586.57015 · doi:10.5802/aif.1066 · numdam:AIF_1986__36_3_207_0 · eudml:74725
[6] A. El Kacimi-Alaoui , V Sergiescu et G. Hector , La cohomologie basique d’un feuilletage riemannien est de dimension finie . Math. Z., 188 (1985), 593-599. · Zbl 0536.57013 · doi:10.1007/BF01161658 · eudml:173561
[7] E. Ghys , Feuilletages riemanniens sur les variétés simplement connexes . Ann. Inst. Fourier, 34, 4 (1984), 203-223. · Zbl 0525.57024 · doi:10.5802/aif.994 · numdam:AIF_1984__34_4_203_0 · eudml:74656
[8] S. Goldberg , Curvature and Homology . Dover Publications, Inc., New-York. · Zbl 0962.53001
[9] P. Griffiths , and J. Harris , Principles of algebraic Geometry. Pure and Applied Mathematics - Interscience Series of Texts . · Zbl 0408.14001
[10] A. Haefliger , Pseudo-groups of local isometries . In Differential Geometry, Santiago de Compostela , Sept. 1984, 174-197. L. Cordero editor, Research notes 131, Pitman (1985). · Zbl 0656.58042
[11] F. Kamber and P. Tondeur , Foliated Bundles and Characteristic classes . Lecture Notes in Math., no. 493, Springer-Verlag (1975). · Zbl 0308.57011 · doi:10.1007/BFb0081558
[12] F. Kamber , and P. Tondeur , Hodge de Rham theory for Riemannian foliations . Math. Ann. 277, 415-431 (1987). · Zbl 0637.53043 · doi:10.1007/BF01458323 · eudml:164257
[13] C. Lazarov , An Index Theorem for foliations . Illinois J. of Math. Vol. 30 no. 1 (1986). · Zbl 0592.58049
[14] P. Molino , Géométrie globale des feuilletages riemanniens . Pro. Kon. Neder. Akad., Ser. A, 1, 85 (1982), 45-76. · Zbl 0516.57016
[15] R.S. Palais , Seminar on the Atiyah-Singer Index Theorem . Ann. of Math. Studies no. 57, Princeton University Press (1965). · Zbl 0137.17002 · doi:10.1515/9781400882045
[16] B.L. Reinhart , Harmonic integrals on almost product manifolds . Trans. AMS, 88 (1958), 243-276. · Zbl 0081.31602 · doi:10.2307/1993250
[17] B.L. Reinhart , Foliated manifold with bundle-like metric . Ann. of Math., 69 (1959), p. 119-132. · Zbl 0122.16604 · doi:10.2307/1970097 · www.jstor.org
[18] M. Saralegui , The Euler Class for flow of isometries . Research Notes in Math 131 (1985) edited by L.A. Cordero. · Zbl 0651.57018
[19] Seminaire Palaiseau , Première classe de Chern et courbure de Ricci: preuve de la conjecture de Calabi - Astérisque no. 58 (1978). · Zbl 0397.35028
[20] J.P. Serre , Un théorème de dualité . Comment. Math. Helv., 29, (1955), 9-26. · Zbl 0067.16101 · doi:10.1007/BF02564268 · eudml:139092
[21] R.C. Wells , Differential Analysis on Complex Manifolds . G. T.M. no. 65, Springer-Verlag (1979). · Zbl 0435.32004
[22] S.T. Yau , On the Ricci curvature of a compact Kählerian manifold and the complex Monge-Ampère equation . Comm. Pure and Appl. Math. XX VI (1978), 339-411. · Zbl 0369.53059 · doi:10.1002/cpa.3160310304
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.