Hostname: page-component-848d4c4894-p2v8j Total loading time: 0.001 Render date: 2024-05-16T13:54:29.925Z Has data issue: false hasContentIssue false
  • English
  • Français

Twistor theory of manifolds with Grassmannian structures

Published online by Cambridge University Press:  22 January 2016

Yoshinori Machida  and
Hajime Sato
Yoshinori Machida
Affiliation:
Numazu College of Technology, 3600 Ooka Numazu-shi, Shizuoka, 410-8501, Japan, nachida@la.numazu-ct.ac.jp
Hajime Sato
Affiliation:
Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya, 464-8602, Japanhsato@math.nagoya-u.ac.jp
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

As a generalization of the conformal structure of type (2, 2), we study Grassmannian structures of type (n, m) for n, m ≥ 2. We develop their twistor theory by considering the complete integrability of the associated null distributions. The integrability corresponds to global solutions of the geometric structures.

A Grassmannian structure of type (n, m) on a manifold M is, by definition, an isomorphism from the tangent bundle TM of M to the tensor product V ⊗ W of two vector bundles V and W with rank n and m over M respectively. Because of the tensor product structure, we have two null plane bundles with fibres Pm-1(ℝ) and Pn-1(ℝ) over M. The tautological distribution is defined on each two bundles by a connection. We relate the integrability condition to the half flatness of the Grassmannian structures. Tanaka’s normal Cartan connections are fully used and the Spencer cohomology groups of graded Lie algebras play a fundamental role.

Besides the integrability conditions corrsponding to the twistor theory, the lifting theorems and the reduction theorems are derived. We also study twistor diagrams under Weyl connections.

Keywords

53C2853C15
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 160 , 2000 , pp. 17 - 102
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2000

References

[A-G1] Akivis, M. A. and Goldberg, V. V., Conformal differential geometry and its gen eralizations, John Wiley and Sons, Inc., New York, 1996.Google Scholar
[A-G2] Akivis, M. A. and Goldberg, V. V., On the theory of almost Grassmann structures, in New developments in differential geometry (Szenthe, J., ed.), Kluwer Academic Publishers, Dordrecht, Boston, London (1998), pp. 1–37.Google Scholar
[A-G3] Akivis, M. A. and Goldberg, V. V., Conformal and Grassmann structures, Differential Geom. Appl., 8 (1998), 177–203.CrossRefGoogle Scholar
[A-G4] Akivis, M. A. and Goldberg, V. V., Semiintegrable almost Grassmann structures, Differential Geom. Appl., 10 (1999), 257–294.CrossRefGoogle Scholar
[B] Besse, A. L., Manifolds all of whose geodesics are closed, Springer-Verlag, Berlin, Heidelberg, New York, 1978.Google Scholar
[B-E] Bailey, T. N. and Eastwood, M. G., Complex paraconformal manifolds — their differential geometry and twistor theory, Forum Math., 3 (1991), 61–103.CrossRefGoogle Scholar
[G] Goncharov, A. B., Generalized conformal structures on manifolds, Selecta Math. Sov., 6 (1987), 307–340.Google Scholar
[Ha] Hangan, T., Geómétrie diffärentielle Grassmannienne, Rev. Roum. Math. Pures Appl., 11 (1966), 519–531.Google Scholar
[He] Helgason, S., Differential geometry, Lie groups, and symmetric spaces, Academic Press, New York, London, 1978.Google Scholar
[Hi] Hitchin, N. J., Complex manifolds and Einstein’s equations, in Twistor geometry and non-linear systems (Doebner, H. D., Palev, T. D., eds.), Lecture Notes in Math. 970, Springer-Verlag, Berlin, Heidelberg, New York (1982), pp. 73–99.Google Scholar
[I] Ishihara, T., On tensor-product structures and Grassmannian structures, J. Math. Tokushima Univ., 4 (1970), 1–17.Google Scholar
[J-T] Jones, P. E. and Tod, K. P., Minitwistor spaces and Einstein-Weyl spaces, Class. Quantum. Grav., 2 (1985), 565–577.Google Scholar
[Ka] Kaneyuki, S., On the subalgebras g 0 and g ev of semisimple graded Lie algebras, J. Math. Soc. Japan., 45 (1993), 1–19.CrossRefGoogle Scholar
[Ko] Kobayashi, S., Transformation groups in differential geometry, Springer-Verlag, Berlin, Heidelberg, New York, 1972.Google Scholar
[K-M] Kamada, H. and Machida, Y., Self-duality of metrics of type (2, 2) on four-dimensional manifolds, Tohoku Math. J., 49 (1997), 259–275.Google Scholar
[K-Na] Kobayashi, S. and Nagano, T., On filterd Lie algebras and geometric structures I, J. Math. Mech., 13 (1964), 875–907.Google Scholar
[K-No] Kobayashi, S. and Nomizu, K., Foundations of differential geometry I, Interscience Publishers, John Wiley and Sons, New York, London, 1963.Google Scholar
[Ma] Manin, Y. I., Gauge field theory and complex geometry, Springer-Verlag, Berlin, Heidelberg, New York, 1988.Google Scholar
[Mi] Mikhailov, Y. I., On the structure of almost Grassmannian manifolds, Soviet Maht., 22 (1978), 54–63.Google Scholar
[Ma-Sa] Machida, Y. and Sato, H., Twistor spaces for real four-dimensional Lorentzian manifolds, Nagoya Math. J., 134 (1994), 107–135.Google Scholar
[Mi-St] Milnor, J. and Stasheff, J.D., Characteristic classes, Princeton Univ., 1974.Google Scholar
[O] Ochiai, T., Geometry associated with semisimple flat homogeneous spaces, Trans. Amer. Math. Soc., 152 (1970), 159–193.CrossRefGoogle Scholar
[P-S] Pedersen, H. and Swann, A., Riemannian submersions, four-manifolds and Einstein-Weyl geometry, Proc. London Math. Soc., 66 (1993), 381–399.Google Scholar
[P-T] Pedersen, H. and Tod, K. P., Three-dimensional Einstein-Weyl geometry, Adv. in Math., 97 (1993), 74–109.Google Scholar
[S] Sternberg, S., Lectures on differential geometry, Prentice-Hall, Inc., New Jersey, 1964.Google Scholar
[S-Y1] Sato, H. and Yamaguchi, K., Lie contact manifods, in Geometry of manifolds (Shiohama, K., ed.), Academic Press, Boston (1989), pp. 191–238.Google Scholar
[S-Y2] Sato, H. and Yamaguchi, K., Lie contact manifolds II, Math. Ann., 297 (1993), 33–57.Google Scholar
[Ta] Takeuchi, M., A remark on Lie contact structures, Science Report Coll. Ge. Ed. Osaka Univ., 42 (1993), 29–37.Google Scholar
[To] Tod, K. P., Compact 3-dimensional Einstein-Weyl structures, J. London Math. Soc., 45 (1992), 341–351.CrossRefGoogle Scholar
[T1] Tanaka, N., On the equivalence problems associated with simple graded Lie algebras, Hokkaido Math. J., 8 (1979), 23–84.CrossRefGoogle Scholar
[T2] Tanaka, N., On affine symmetric spaces and the automorphism groups of product manifolds, Hokkaido Math. J., 14 (1985), 277–351.Google Scholar
[T3] Tanaka, N., On geometric theory of systems of ordinary differential equations, Lectures in Colloq. on Diff. Geom. at Sendai, August, 1989.Google Scholar
[W] Wells, R. O. Jr., Complex geometry in mathematical physics, Presses de l’Université de Montréal, Montréal, 1982.Google Scholar
[W-W] Ward, R. S. and Wells, R. O. Jr., Twistor geometry and field theory, Cambridge Univ. Press, 1990.CrossRefGoogle Scholar
[Y] Yamaguchi, K., Differential systems associated with simple graded Lie algebras, Advanced Studies in Pure Math., 22 (1993), 413–494.Google Scholar
[Y-I] Yano, K. and Ishihara, S., Tangent and cotangent bundles, Marcel Dekker, New york, 1973.Google Scholar