Nagoya Mathematical Journal

Twistor theory of manifolds with Grassmannian structures

Yoshinori Machida and Hajime Sato

Full-text: Open access

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

Permanent link to this document
https://projecteuclid.org/euclid.nmj/1114631499

Mathematical Reviews number (MathSciNet)
MR1804138

Zentralblatt MATH identifier
1039.53055

Subjects
Primary: 53C28: Twistor methods [See also 32L25]
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


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