Nagoya Mathematical Journal

Twistor theory of manifolds with Grassmannian structures

Yoshinori Machida and Hajime Sato

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As a generalization of the conformal structure of type $(2, 2)$, we study Grassmannian structures of type $(n, m)$ for $n, m \geq 2$. 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

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

First available in Project Euclid: 27 April 2005

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C28: Twistor methods [See also 32L25]
Secondary: 53C10: $G$-structures


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

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