Sub-bundles of the complexified tangent bundle

Jacobowitz, Howard; Mendoza, Gerardo

doi:10.1090/S0002-9947-03-03350-6

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Browser Compatibility Info Help
ISSN 1088-6850(online) ISSN 0002-9947(print)

Sub-bundles of the complexified tangent bundle

Authors: Howard Jacobowitz and Gerardo Mendoza
Journal: Trans. Amer. Math. Soc. 355 (2003), 4201-4222
MSC (2000): Primary 57R22, 58J10; Secondary 35F05, 35N10
Published electronically: June 10, 2003
MathSciNet review: 1990583
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study embeddings of complex vector bundles, especially line bundles, in the complexification of the tangent bundle of a manifold. The aim is to understand implications of properties of interest in partial differential equations.

1. Aldo Andreotti and C. Denson Hill, Complex characteristic coordinates and tangential Cauchy-Riemann equations, Ann. Scuola Norm. Sup. Pisa (3) 26 (1972), 299–324. MR 0460724 (57 #717)
2. M. S. Baouendi, C. H. Chang, and F. Trèves, Microlocal hypo-analyticity and extension of CR functions, J. Differential Geom. 18 (1983), no. 3, 331–391. MR 723811 (85h:32030)
3. Raoul Bott, On a topological obstruction to integrability, Global Analysis (Proc. Sympos. Pure Math., Vol. XVI, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970, pp. 127–131. MR 0266248 (42 #1155)
4. Raoul Bott and Loring W. Tu, Differential forms in algebraic topology, Graduate Texts in Mathematics, vol. 82, Springer-Verlag, New York, 1982. MR 658304 (83i:57016)
5. Marco Brunella, On transversely holomorphic flows. I, Invent. Math. 126 (1996), no. 2, 265–279. MR 1411132 (97j:58121), http://dx.doi.org/10.1007/s002220050098
6. Martin Čadek and Jiří Vanžura, On complex structures in 8-dimensional vector bundles, Manuscripta Math. 95 (1998), no. 3, 323–330. MR 1612074 (98m:57034), http://dx.doi.org/10.1007/s002290050032
7. Martin Čadek and Jiří Vanžura, On 2-distributions in 8-dimensional vector bundles over 8-complexes, Colloq. Math. 70 (1996), no. 1, 25–40. MR 1373279 (96k:57021)
8. Shiing-shen Chern, An elementary proof of the existence of isothermal parameters on a surface, Proc. Amer. Math. Soc. 6 (1955), 771–782. MR 0074856 (17,657g), http://dx.doi.org/10.1090/S0002-9939-1955-0074856-1
9. Étienne Ghys, On transversely holomorphic flows. II, Invent. Math. 126 (1996), no. 2, 281–286. MR 1411133 (97j:58122), http://dx.doi.org/10.1007/s002220050099
10. Terence Heaps, Almost complex structures on eight- and ten-dimensional manifolds, Topology 9 (1970), 111–119. MR 0264692 (41 #9283)
11. Friedrich Hirzebruch and Heinz Hopf, Felder von Flächenelementen in 4-dimensionalen Mannigfaltigkeiten, Math. Ann. 136 (1958), 156–172 (German). MR 0100844 (20 #7272)
12. C. C. Hsiung, Almost complex and complex structures, Series in Pure Mathematics, vol. 20, World Scientific Publishing Co., Singapore, 1995. MR 1356223 (96j:53033)
13. Dale Husemoller, Fibre bundles, 3rd ed., Graduate Texts in Mathematics, vol. 20, Springer-Verlag, New York, 1994. MR 1249482 (94k:55001)
14. Howard Jacobowitz, Global Mizohata structures, J. Geom. Anal. 3 (1993), no. 2, 153–193. MR 1214706 (94c:58190), http://dx.doi.org/10.1007/BF02921581
15. Howard Jacobowitz, Transversely holomorphic foliations and CR structures, Mat. Contemp. 18 (2000), 175–194. VI Workshop on Partial Differential Equations, Part I (Rio de Janeiro, 1999). MR 1812870 (2002f:32061)
16. Michel A. Kervaire, A note on obstructions and characteristic classes, Amer. J. Math. 81 (1959), 773–784. MR 0107863 (21 #6585)
17. Proceedings of Liverpool Singularities— Symposium. I (1969/1970)., Edited by C. T. C. Wall. Lecture Notes in Mathematics, Vol. 192, Springer-Verlag, Berlin, 1971. MR 0278325 (43 #4055)
18. Robert Macpherson, Generic vector bundle maps, Dynamical systems (Proc. Sympos., Univ. Bahia, Salvador, 1971) Academic Press, New York, 1973, pp. 165–175. MR 0339199 (49 #3962)
19. Maria Hermínia de Paula Leite Mello, Two-plane sub-bundles of nonorientable real vector-bundles, Manuscripta Math. 57 (1987), no. 3, 263–280. MR 873468 (88c:55020), http://dx.doi.org/10.1007/BF01437484
20. W. S. Massey, On the Stiefel-Whitney classes of a manifold, Amer. J. Math. 82 (1960), 92–102. MR 0111053 (22 #1918)
21. G. A. Mendoza, Elliptic structures, (São Carlos, 1995) Contemp. Math., vol. 205, Amer. Math. Soc., Providence, RI, 1997, pp. 219–234. MR 1447226 (98c:32041), http://dx.doi.org/10.1090/conm/205/02665
22. John W. Milnor and James D. Stasheff, Characteristic classes, Princeton University Press, Princeton, N. J., 1974. Annals of Mathematics Studies, No. 76. MR 0440554 (55 #13428)
23. Franklin P. Peterson, Some remarks on Chern classes, Ann. of Math. (2) 69 (1959), 414–420. MR 0102807 (21 #1593)
24. H. Shulman, Secondary obstructions to foliations, Topology 13 (1974), 177–183. MR 0345116 (49 #9855)
25. Norman Steenrod, The Topology of Fibre Bundles, Princeton Mathematical Series, vol. 14, Princeton University Press, Princeton, N. J., 1951. MR 0039258 (12,522b)
26. Emery Thomas, Fields of tangent 2-planes on even-dimensional manifolds, Ann. of Math. (2) 86 (1967), 349–361. MR 0212834 (35 #3699)
27. Emery Thomas, Fields of tangent 𝑘-planes on manifolds, Invent. Math. 3 (1967), 334–347. MR 0217814 (36 #903)
28. Emery Thomas, Complex structures on real vector bundles, Amer. J. Math. 89 (1967), 887–908. MR 0220310 (36 #3375)
29. Emery Thomas, Secondary obstructions to integrability, Dynamical systems (Proc. Sympos., Univ. Bahia, Salvador, 1971) Academic Press, New York, 1973, pp. 655–661. MR 0343293 (49 #8035)
30. François Trèves, Hypo-analytic structures, Princeton Mathematical Series, vol. 40, Princeton University Press, Princeton, NJ, 1992. Local theory. MR 1200459 (94e:35014)
31. Vladimir Turaev, Torsion invariants of 𝑆𝑝𝑖𝑛^{𝑐}-structures on 3-manifolds, Math. Res. Lett. 4 (1997), no. 5, 679–695. MR 1484699 (98k:57038), http://dx.doi.org/10.4310/MRL.1997.v4.n5.a6
32. John W. Wood, Foliations on 3-manifolds, Ann. of Math. (2) 89 (1969), 336–358. MR 0248873 (40 #2123)