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Twistor theory of manifolds with Grassmannian structures. (English) Zbl 1039.53055
As a generalization of the conformal structure of type ($$2,2$$), the authors study Grassmanian structures of type ($$n,m$$) for $$n,m\geq 2$$. They 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 an isomorphism from the tangent bundle $$T(M)$$ of $$M$$ to the tensor product $$V\otimes W$$ of two vector bundles $$V$$ and $$W$$ with rank $$n$$ and $$m$$ over $$M$$ respectively. Typical examples are Grassmannian manifolds (homogeneous models) which are the flat models. The authors give some other examples and show that, in the four-dimensional case, the notion of Grassmannian structure of type ($$2,2$$) is equivalent to the notion of conformal structure of type ($$2,2$$). They also describe a topological obstruction to the existence of a Grassmannian structure of type ($$n,2$$); as consequence the sphere $$S^{2n}$$ and the quaternionic projective space $$P^m(\mathbb{H})$$ ($$n=2m$$) admit no Grassmannian structure of type ($$n,2$$).
Given a Grassmannian structure of type ($$n,m$$) on a manifold $$M$$, the set $$F_L$$ of all tangent $$n$$-planes of the form $$\{v\otimes w \mid v\in V_x\}$$ where $$w\in W_x$$, $$w\neq 0$$, $$x\in M$$ is an $$\mathbb{R} P^{m-1}$$-bundle over $$M$$. This bundle is associated with the principal bundle $$Q$$ of frames of second order with structure group the subgroup of $$SL(n+m,\mathbb{R})$$ that leaves $$\mathbb{R}^m$$ invariant. The bundle $$Q$$ admits a unique normal Cartan connection $$\omega$$ in the sense of N. Tanaka [Hokkaido Math. J. 8, 23–84 (1979; Zbl 0409.17013)]. By this connection the notion of half flatness for the Grassmannian structure of type ($$n,m$$) is defined. One can define an $$n$$-dimensional tautological distribution $$D_L$$ on $$F_L$$. Similarly, the set $$F_R$$ of $$m$$-planes $$\{v\otimes w \mid w\in W_x\}$$, $$v\in V_x$$, $$v\neq 0$$, $$x\in M$$, is an $$\mathbb{R} P^{n-1}$$-bundle which admits a tautological distribution $$D_R$$. The authors express the integrability condition for $$D_L$$ (resp. $$D_R$$) in terms of the curvature of $$\omega$$ involving Spencer cohomology groups of graded Lie algebras. They prove that the distribution $$D_L$$ on $$F_L$$ over $$M$$ is completely integrable if and only if the Grassmannian structure on $$M$$ is right-half flat and they give a non-flat half-flat example. In the same way the authors give the condition for $$D_R$$ to be completely integrable, i.e. left-half flatness. They also discuss the double fibration of Grassmannian structure, that is, the twistor diagram of the homogeneous model and interpret it in terms of the Dynkin diagrams. Then, the authors consider Weyl connections associated with conformal structures instead of normal Cartan connections associated with Grassmannian structure and show that a Weyl structure with constant curvature induces a “right-half flat” Grassmannian structure of type ($$n,2$$) on the orbit space of its geodesic flow.

##### MSC:
 53C28 Twistor methods in differential geometry 53C10 $$G$$-structures
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