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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, [email protected]
Hajime Sato
Affiliation:
Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya, 464-8602, [email protected]
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Abstract

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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

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