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2000 Twistor theory of manifolds with Grassmannian structures
Yoshinori Machida, Hajime Sato
Nagoya Math. J. 160(none): 17-102 (2000).

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.

Citation

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Yoshinori Machida. Hajime Sato. "Twistor theory of manifolds with Grassmannian structures." Nagoya Math. J. 160 17 - 102, 2000.

Information

Published: 2000
First available in Project Euclid: 27 April 2005

zbMATH: 1039.53055
MathSciNet: MR1804138

Subjects:
Primary: 53C28
Secondary: 53C10

Rights: Copyright © 2000 Editorial Board, Nagoya Mathematical Journal

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