## Nagoya Mathematical Journal

### Twistor theory of manifolds with Grassmannian structures

#### Abstract

As a generalization of the conformal structure of type $(2, 2)$, we study Grassmannian structures of type $(n, m)$ for $n, m \geq 2$. We develop their twistor theory by considerin 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 $V \otimes W$ 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 $P^{m-1}(\mathbb{R})$ and $P^{n-1}(\mathbb{R})$ 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 corr[e]sponding to the twistor theory, the lifting theorems and the reduction theorems are derived. We also study twistor diagrams under Weyl connections.

#### Article information

Source
Nagoya Math. J., Volume 160 (2000), 17-102.

Dates
First available in Project Euclid: 27 April 2005

https://projecteuclid.org/euclid.nmj/1114631499

Mathematical Reviews number (MathSciNet)
MR1804138

Zentralblatt MATH identifier
1039.53055

Subjects
Secondary: 53C10: $G$-structures

#### Citation

Machida, Yoshinori; Sato, Hajime. Twistor theory of manifolds with Grassmannian structures. Nagoya Math. J. 160 (2000), 17--102. https://projecteuclid.org/euclid.nmj/1114631499

#### References

• M. A. Akivis and V. V. Goldberg, Conformal differential geometry and its generalizations, John Wiley and Sons, Inc., New York (1996).
• ––––, On the theory of almost Grassmann structures , in New developments in differential geometry (J. Szenthe, ed.), Kluwer Academic Publishers, Dordrecht, Boston, London (1998), 1–37.
• ––––, Conformal and Grassmann structures , Differential Geom. Appl., 8 (1998), 177–203.
• ––––, Semiintegrable almost Grassmann structures , Differential Geom. Appl., 10 (1999), 257–294.
• A. L. Besse, Manifolds all of whose geodesics are closed, Springer-Verlag, Berlin, Heidelberg, New York (1978).
• T. N. Bailey and M. G. Eastwood, Complex paraconformal manifolds –- their differential geometry and twistor theory , Forum Math., 3 (1991), 61–103.
• A. B. Goncharov, Generalized conformal structures on manifolds , Selecta Math. Sov., 6 (1987), 307–340.
• T. Hangan, Geómétrie différentielle Grassmannienne , Rev. Roum. Math. Pures Appl., 11 (1966), 519–531.
• S. Helgason, Differential geometry, Lie groups, and symmetric spaces, Academic Press, New York, London (1978).
• N. J. Hitchin, Complex manifolds and Einstein's equations , in Twistor geometry and non-linear systems (H. D. Doebner, T. D. Palev, eds.), Lecture Notes in Math. 970, Springer-Verlag, Berlin, Heidelberg, New York (1982), 73–99.
• T. Ishihara, On tensor-product structures and Grassmannian structures , J. Math. Tokushima Univ., 4 (1970), 1–17.
• P. E. Jones and K. P. Tod, Minitwistor spaces and Einstein-Weyl spaces , Class. Quantum. Grav., 2 (1985), 565–577.
• S. Kaneyuki, On the subalgebras ${\smallfrak g}_{0}$ and ${\smallfrak g}_{\it ev}$ of semisimple graded Lie algebras , J. Math. Soc. Japan., 45 (1993), 1–19.
• S. Kobayashi, Transformation groups in differential geometry, Springer-Verlag, Berlin, Heidelberg, New York (1972).
• H. Kamada and Y. Machida, Self-duality of metrics of type $(2, 2)$ on four-dimensional manifolds , Tohoku Math. J., 49 (1997), 259–275.
• S. Kobayashi and T. Nagano, On filterd Lie algebras and geometric structures I , J. Math. Mech., 13 (1964), 875–907.
• S. Kobayashi and K. Nomizu, Foundations of differential geometry I, Interscience Publishers, John Wiley and Sons, New York, London (1963).
• Y. I. Manin, Gauge field theory and complex geometry, Springer-Verlag, Berlin, Heidelberg, New York (1988).
• Y. I. Mikhailov, On the structure of almost Grassmannian manifolds , Soviet Maht., 22 (1978), 54–63.
• Y. Machida and H. Sato, Twistor spaces for real four-dimensional Lorentzian manifolds , Nagoya Math. J., 134 (1994), 107–135.
• J. Milnor and J.D. Stasheff, Characteristic classes, Princeton Univ. (1974).
• T. Ochiai, Geometry associated with semisimple flat homogeneous spaces , Trans. Amer. Math. Soc., 152 (1970), 159–193.
• H. Pedersen and A. Swann, Riemannian submersions, four-manifolds and Einstein-Weyl geometry , Proc. London Math. Soc., 66 (1993), 381–399.
• H. Pedersen and K. P. Tod, Three-dimensional Einstein-Weyl geometry , Adv. in Math., 97 (1993), 74–109.
• S. Sternberg, Lectures on differential geometry, Prentice-Hall, Inc., New Jersey (1964).
• H. Sato and K. Yamaguchi, Lie contact manifods , in Geometry of manifolds (K. Shiohama, ed.), Academic Press, Boston (1989), 191–238.
• ––––, Lie contact manifolds II , Math. Ann., 297 (1993), 33–57.
• M. Takeuchi, A remark on Lie contact structures , Science Report Coll. Ge. Ed. Osaka Univ., 42 (1993), 29–37.
• K. P. Tod, Compact $3$-dimensional Einstein-Weyl structures , J. London Math. Soc., 45 (1992), 341–351.
• N. Tanaka, On the equivalence problems associated with simple graded Lie algebras , Hokkaido Math. J., 8 (1979), 23–84.
• ––––, On affine symmetric spaces and the automorphism groups of product manifolds , Hokkaido Math. J., 14 (1985), 277–351.
• ––––, On geometric theory of systems of ordinary differential equations, Lectures in Colloq. on Diff. Geom. at Sendai, August (1989).
• R. O. Wells, Jr., Complex geometry in mathematical physics, Presses de l'Université de Montréal, Montréal (1982).
• R. S. Ward and R. O. Wells, Jr., Twistor geometry and field theory, Cambridge Univ. Press (1990).
• K. Yamaguchi, Differential systems associated with simple graded Lie algebras , Advanced Studies in Pure Math., 22 (1993), 413–494.
• K. Yano and S. Ishihara, Tangent and cotangent bundles, Marcel Dekker, New york (1973).