Un survol sur la théorie de Hodge-de Rham des variétés lisses et singulières
DOI:
https://doi.org/10.22199/S07160917.1993.0002.00001Keywords:
Topología algebraica, Grupos, Clavos, Espacios vectoriales, CálculoAbstract
L 'objet de ce papier est de faire un léger survol sur la cohomologie des formes différentielles : de de Rham et sa représentation par les formes harmoniques dans le cas compact lisse, des formes L2 dans le cas singulier et enfin les formes basiques et leur théorie de Hodge pour le cas d 'un feuilletage riemannien sur une variété compacte.
References
[AV]. Andreotti, A. and Vesentini, E., Carleman estimates for the Laplace-
Beltrami equations, Pub. Math. I.H.E.S. (1965), 81-130.
[Ati]. Atiyah, M., Elliptic operators, discrete groups and von Neumann algebras, Asterisque 32-33 (1976), 43-72.
[APS]. Atiyah, M., Patodi, V. and Singer, I., Spectral assymetry and Riemannian geometry, I, JI and III, Math. Pro. Cam. Phil. Soc., 77 (1975), 43-69, 78 (1976), 405-432 and (79 (1976)), 315-330.
[Bai]. Bailly, W., The decomposition theorem for V -manifolds, Amer. J. of Math., 78 (1956), 862-888.
[BB]. Belliart, M. et Birembaux, 0., Actions localement libres de groupes résolubles, Prépublication Université de Valenciennes (1993).
[Cam]. Camacho, C. and Neto, A., Geometric Theory of Foliations, Birkhäuser.
[Che]. Cheeger, J., On the Hodge theory of Riemannian Pseudomanifolds, Proc. of Symp. in Pure Math., 36 (1980), 91-146.
[CGM]. Cheeger, J., Goresky, M. and Macpherson, R., L2 -Cohomology and Intersection homology, Ann. of Math. Studies, 102 (1982), 303-340.
[CM]. Connes, A. and Moscovici, H., L2 -index theorem for homogeneous spaces of Lie groups, Ann. of Math., 115, (1982), 291-330.
[Deb]. DeBaun, D., L2 -cohomology of non compact surfaces, Trans. of AMS (1984), 543-565.
[Dod1]. Dodziuk, J., Vanishing theorems for L2 -harmonic forms, Volume in memory of V.K. Patodi, Tata Institute of Fundamental Research (Bombay) (1983).
[Dod2]. Dodziuk, J., de Rham-Hodge theorry for L2 -cohomology of infinite coverings, Topology, 16 (1977), 157-165.
[Dod3]. Dodziuk, J., Sobolev spaces of differential forms and de Rham-Hodge isomorphism, J. of Diff. Geo., 16 (1981), 63-73.
[Dod4]. Dodziuk, J., L2 - harmonic forms on rotationally symetric Riemannian manifolds, Proc. Amer. Math. Soc., 77 (1979), 395-400.
[DF]. Donneley, H. and Fefferman, C., L2 -Cohomology and Index Theorem for the Bergman metric, Ann. of Math., 118 (1983), 593-618.
[Elk1]. El Kacimi Alaoui, A., Opérateurs transversalement elliptiques sur un feuilletage riemannien et applications, Composito Mathematica, 73 (1990), 57-106.
[Elk2]. El Kacimi Alaoui, A., Dualité pour les feuilletages transversalement holomorphes, Manuscripta Math. 58 (1987), 417-443.
[EH]. El Kacimi Alaoui. A et Héctor G. Decomposition de Hedge basique pour un feuilletages riemannien, Ann. Inst. Grenoble, 36, 3 (1986). 207-227.
[ESH]. El Kacimi Alaoui, A., Sergiescu, V et Héctor G. La cohomologie basique d'un feuilletage riemannien est de dimension finie Math Zei., 188 (1985), 593-599.
[EN1] El Kacimi Alaoui, A. and Nicolau, M., On the topological invariance of the basic cohomology, Math. Ann., 295 (1993), 627-634.
[EN2]. El Kacimi Alaoui, A. and Nicolau, M., Structures géométriques invariantes et feuilletages de Lie, Indagationes Mathematicae, New series, Vol 1, No. 3 (1990), 323-334.
[EN3]. El Kacimi Alaoui, A. and Nicolau, M., A class of C?-stable foliations, Ergod. Th. & Dynam. Sys. 13 (1993), 1-8.
[Fed]. Fedida, E., Sur les feuilletages de Lie, CRAS de Paris, 272 (1971),
999-1002.
[Gaf1]. Gaffney, M., A special Stokes' Theorem for Complete Riemanntan manifolds, Ann. of Math., Vol. 60 (1954), 140-145.
[Ghy1]. Ghys, E, Un feuilletage analytique dont la cohomologie basique est de dimension infinie, Pub. IRMA- Lille, Vol. VII, Fase. 1 (1985).
[Ghy2]. Ghys, E, Actions localement libres du groupe affine, Inventiones Math., 82 (1985), 479-526.
[Ghy3]. Ghys, E., Feuilletages riemanniens sur les variétés simplement connexes, Ann. Inst. Fourier, 34, 4 (1984), 203-223.
[God]. Godbillon, C., Feuilletages. Etudes géométriques, Birkhäuser (1991).
[Gol]. Goldberg, S., Curvature and cohomology, Collection Dover.
[Hae]. Haefliger, A., Leaf closures in Riemannian foliations, A fête on Topology. Papers dedicated to Itiro Tamura; Academic Press (1988).
[HH] Hector, G. and Hirsch, U., Introduction to the Geometry of Foliations, Parts A and B. Aspects of Mathematics, Vieweg.
[Hod]. Hodge, W. D., The Theory and Applications of Harmonic Integrals, Cambridge University Press, New York (1952).
[Kaw]. Kawasaki, T., The index of elliptic operators over V -manifolds, Nagoya Math J , Vol. 84 (1981), 135-157.
[Kit1]. Kitahara, H.. Nonexistence of nontrivial ?”-harmonic 1-forms on a complete fohated Riemannian manifold, Trans. of AMS, 262, No. 2 (1980), 429-435 .
[Kit2]. Kitahara, H., Remarks on square integrable basic cohomology spaces on a foliated Riemannian foliation, Kodai Math . J., 2 (1979), 187-193.
[Kod] Kodaira, K., Harmonic fields in Riemannian manifolds, Annals of Math., 50 (1949), 587-665.
[Mac]. MacPherson, R., Global questions in the topology of singular spaces, ICM, Warsaw (1983).
[Max]. Maxa. J., Duality and minimality in Riemannian foliations, Comment. Math. Helvetici, 67 (1992), 17-27.
[Mol]. Molino, P., Riemannian Foliations, Progress in Math. Vol. 73, Birkhäuser.
[ME]. Morrey, C. and Eells, J., A variational method in the theory of harmonic integrals, I, Ann. of Math., Vol. 63 (1956), 91-128.
[Nag1]. Nagase, M., L2 -Cohomology and Intersection cohomology of stratified spaces, Duke Math. J., 50 (1983), 329-368.
[Nag2]. Nagase, M., Sheaf theoretic L2 -cohomology, Advanced Stu. Pure Math. No. 8; Nort.h Holland (1987), 273-279.
[Rha]. de Rham, G., Variétés différentiables, Actualités scientifiques et Industrielles.
[Ren1]. Reinhart, B., Foliated manifolds with bundle-like metrics, Ann. Of Math., 69 (1959), 119-132.
[Ren2]. Reinhart, B., Differential Geometry of Foliations, Ergeb. der Mathematik (1983).
[Ren3]. Reinhart, B., Harmonic integrals on foliated manifolds, Amer. J. of Math., 81 (1959), 529-586.
[Rum]. Rummler, H., Quelques notions simples en géométrie riemannienne et leurs applications aux feuilletages compacts, Comment.
Math. Helv., 54 (1979), 224-239.
[Sag1]. Saralegi, M., Un théorème de dualité entre l'homologie d'intersection et la cohomologie L2, Thèse Université de Lille I (1988).
[Sag2J. Saralegi, M., The Euler class for flows of isometries, Research Notes in Math. 131. Edited by L. Cordem (1985).
[Sar]. Sarkaria, K.S., Non degenerescence of some spectral sequences, Ann. Inst. Fourier de Grenoble, 34, 1 (1984), 39-46.
[Sch]. Schwarz, G.S., On the de Rham cohomology of the leaf space of a foliation, Topology 13 (1974), 185-187.
[Ser1]. Sergiescu, V., Cohomologie basique et dualité des feuilletages riemanniens, Ann. Inst. Fourier, Grenoble 35, 3 (1985), 137-158.
[War]. Warner, W., Foundations of Differentiable Manifolds and Líe Groups, GTM 94.
[Wel]. Wells, R.O., Differential Analysis on Complex Manifolds, GTM 65 (1979).
[Zuc]. Zucker, S., Hodge theory with degenerating coefficients : L2 -Cohomology in the Poincaré Metric, Ann. of Math., 109 (1979), 415-476.
Beltrami equations, Pub. Math. I.H.E.S. (1965), 81-130.
[Ati]. Atiyah, M., Elliptic operators, discrete groups and von Neumann algebras, Asterisque 32-33 (1976), 43-72.
[APS]. Atiyah, M., Patodi, V. and Singer, I., Spectral assymetry and Riemannian geometry, I, JI and III, Math. Pro. Cam. Phil. Soc., 77 (1975), 43-69, 78 (1976), 405-432 and (79 (1976)), 315-330.
[Bai]. Bailly, W., The decomposition theorem for V -manifolds, Amer. J. of Math., 78 (1956), 862-888.
[BB]. Belliart, M. et Birembaux, 0., Actions localement libres de groupes résolubles, Prépublication Université de Valenciennes (1993).
[Cam]. Camacho, C. and Neto, A., Geometric Theory of Foliations, Birkhäuser.
[Che]. Cheeger, J., On the Hodge theory of Riemannian Pseudomanifolds, Proc. of Symp. in Pure Math., 36 (1980), 91-146.
[CGM]. Cheeger, J., Goresky, M. and Macpherson, R., L2 -Cohomology and Intersection homology, Ann. of Math. Studies, 102 (1982), 303-340.
[CM]. Connes, A. and Moscovici, H., L2 -index theorem for homogeneous spaces of Lie groups, Ann. of Math., 115, (1982), 291-330.
[Deb]. DeBaun, D., L2 -cohomology of non compact surfaces, Trans. of AMS (1984), 543-565.
[Dod1]. Dodziuk, J., Vanishing theorems for L2 -harmonic forms, Volume in memory of V.K. Patodi, Tata Institute of Fundamental Research (Bombay) (1983).
[Dod2]. Dodziuk, J., de Rham-Hodge theorry for L2 -cohomology of infinite coverings, Topology, 16 (1977), 157-165.
[Dod3]. Dodziuk, J., Sobolev spaces of differential forms and de Rham-Hodge isomorphism, J. of Diff. Geo., 16 (1981), 63-73.
[Dod4]. Dodziuk, J., L2 - harmonic forms on rotationally symetric Riemannian manifolds, Proc. Amer. Math. Soc., 77 (1979), 395-400.
[DF]. Donneley, H. and Fefferman, C., L2 -Cohomology and Index Theorem for the Bergman metric, Ann. of Math., 118 (1983), 593-618.
[Elk1]. El Kacimi Alaoui, A., Opérateurs transversalement elliptiques sur un feuilletage riemannien et applications, Composito Mathematica, 73 (1990), 57-106.
[Elk2]. El Kacimi Alaoui, A., Dualité pour les feuilletages transversalement holomorphes, Manuscripta Math. 58 (1987), 417-443.
[EH]. El Kacimi Alaoui. A et Héctor G. Decomposition de Hedge basique pour un feuilletages riemannien, Ann. Inst. Grenoble, 36, 3 (1986). 207-227.
[ESH]. El Kacimi Alaoui, A., Sergiescu, V et Héctor G. La cohomologie basique d'un feuilletage riemannien est de dimension finie Math Zei., 188 (1985), 593-599.
[EN1] El Kacimi Alaoui, A. and Nicolau, M., On the topological invariance of the basic cohomology, Math. Ann., 295 (1993), 627-634.
[EN2]. El Kacimi Alaoui, A. and Nicolau, M., Structures géométriques invariantes et feuilletages de Lie, Indagationes Mathematicae, New series, Vol 1, No. 3 (1990), 323-334.
[EN3]. El Kacimi Alaoui, A. and Nicolau, M., A class of C?-stable foliations, Ergod. Th. & Dynam. Sys. 13 (1993), 1-8.
[Fed]. Fedida, E., Sur les feuilletages de Lie, CRAS de Paris, 272 (1971),
999-1002.
[Gaf1]. Gaffney, M., A special Stokes' Theorem for Complete Riemanntan manifolds, Ann. of Math., Vol. 60 (1954), 140-145.
[Ghy1]. Ghys, E, Un feuilletage analytique dont la cohomologie basique est de dimension infinie, Pub. IRMA- Lille, Vol. VII, Fase. 1 (1985).
[Ghy2]. Ghys, E, Actions localement libres du groupe affine, Inventiones Math., 82 (1985), 479-526.
[Ghy3]. Ghys, E., Feuilletages riemanniens sur les variétés simplement connexes, Ann. Inst. Fourier, 34, 4 (1984), 203-223.
[God]. Godbillon, C., Feuilletages. Etudes géométriques, Birkhäuser (1991).
[Gol]. Goldberg, S., Curvature and cohomology, Collection Dover.
[Hae]. Haefliger, A., Leaf closures in Riemannian foliations, A fête on Topology. Papers dedicated to Itiro Tamura; Academic Press (1988).
[HH] Hector, G. and Hirsch, U., Introduction to the Geometry of Foliations, Parts A and B. Aspects of Mathematics, Vieweg.
[Hod]. Hodge, W. D., The Theory and Applications of Harmonic Integrals, Cambridge University Press, New York (1952).
[Kaw]. Kawasaki, T., The index of elliptic operators over V -manifolds, Nagoya Math J , Vol. 84 (1981), 135-157.
[Kit1]. Kitahara, H.. Nonexistence of nontrivial ?”-harmonic 1-forms on a complete fohated Riemannian manifold, Trans. of AMS, 262, No. 2 (1980), 429-435 .
[Kit2]. Kitahara, H., Remarks on square integrable basic cohomology spaces on a foliated Riemannian foliation, Kodai Math . J., 2 (1979), 187-193.
[Kod] Kodaira, K., Harmonic fields in Riemannian manifolds, Annals of Math., 50 (1949), 587-665.
[Mac]. MacPherson, R., Global questions in the topology of singular spaces, ICM, Warsaw (1983).
[Max]. Maxa. J., Duality and minimality in Riemannian foliations, Comment. Math. Helvetici, 67 (1992), 17-27.
[Mol]. Molino, P., Riemannian Foliations, Progress in Math. Vol. 73, Birkhäuser.
[ME]. Morrey, C. and Eells, J., A variational method in the theory of harmonic integrals, I, Ann. of Math., Vol. 63 (1956), 91-128.
[Nag1]. Nagase, M., L2 -Cohomology and Intersection cohomology of stratified spaces, Duke Math. J., 50 (1983), 329-368.
[Nag2]. Nagase, M., Sheaf theoretic L2 -cohomology, Advanced Stu. Pure Math. No. 8; Nort.h Holland (1987), 273-279.
[Rha]. de Rham, G., Variétés différentiables, Actualités scientifiques et Industrielles.
[Ren1]. Reinhart, B., Foliated manifolds with bundle-like metrics, Ann. Of Math., 69 (1959), 119-132.
[Ren2]. Reinhart, B., Differential Geometry of Foliations, Ergeb. der Mathematik (1983).
[Ren3]. Reinhart, B., Harmonic integrals on foliated manifolds, Amer. J. of Math., 81 (1959), 529-586.
[Rum]. Rummler, H., Quelques notions simples en géométrie riemannienne et leurs applications aux feuilletages compacts, Comment.
Math. Helv., 54 (1979), 224-239.
[Sag1]. Saralegi, M., Un théorème de dualité entre l'homologie d'intersection et la cohomologie L2, Thèse Université de Lille I (1988).
[Sag2J. Saralegi, M., The Euler class for flows of isometries, Research Notes in Math. 131. Edited by L. Cordem (1985).
[Sar]. Sarkaria, K.S., Non degenerescence of some spectral sequences, Ann. Inst. Fourier de Grenoble, 34, 1 (1984), 39-46.
[Sch]. Schwarz, G.S., On the de Rham cohomology of the leaf space of a foliation, Topology 13 (1974), 185-187.
[Ser1]. Sergiescu, V., Cohomologie basique et dualité des feuilletages riemanniens, Ann. Inst. Fourier, Grenoble 35, 3 (1985), 137-158.
[War]. Warner, W., Foundations of Differentiable Manifolds and Líe Groups, GTM 94.
[Wel]. Wells, R.O., Differential Analysis on Complex Manifolds, GTM 65 (1979).
[Zuc]. Zucker, S., Hodge theory with degenerating coefficients : L2 -Cohomology in the Poincaré Metric, Ann. of Math., 109 (1979), 415-476.
Published
2018-04-03
How to Cite
[1]
A. El Kacimi Alaoui, “Un survol sur la théorie de Hodge-de Rham des variétés lisses et singulières”, Proyecciones (Antofagasta, On line), vol. 12, no. 2, pp. 63-118, Apr. 2018.
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