Twistor theory of manifolds with Grassmannian structures.

*(English)*Zbl 1039.53055As 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.

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.

Reviewer: Marian Munteanu (Iaşi)

##### Keywords:

Grassmannian structure; conformal structure; twistor theory; Cartan connection; Weyl connection; Spencer cohomology
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\textit{Y. Machida} and \textit{H. Sato}, Nagoya Math. J. 160, 17--102 (2000; Zbl 1039.53055)

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##### References:

[1] | Rev. Roum. Math. Pures Appl 11 pp 519– (1966) |

[2] | Science Report Coll. Ge. Ed. Osaka Univ 42 pp 29– (1993) |

[3] | Selecta Math. Sov 6 pp 307– (1987) |

[4] | Forum Math 3 pp 61– (1991) |

[5] | Advanced Studies in Pure Math 22 pp 413– (1993) |

[6] | Manifolds all of whose geodesics are closed (1978) · Zbl 0387.53010 |

[7] | Twistor geometry and field theory (1990) |

[8] | DOI: 10.1016/S0926-2245(99)00014-5 · Zbl 0921.53006 |

[9] | Presses de l’Université de Montréal, Montréal (1982) |

[10] | Differential geometry, Lie groups, and symmetric spaces (1978) · Zbl 0451.53038 |

[11] | DOI: 10.1016/S0926-2245(98)00007-2 · Zbl 0924.53025 |

[12] | Lectures in Colloq. on Diff. Geom. at Sendai (1989) |

[13] | New developments in differential geometry pp 1– (1998) |

[14] | DOI: 10.14492/hokmj/1381757644 · Zbl 0585.53044 |

[15] | Conformal differential geometry and its gen eralizations (1996) |

[16] | DOI: 10.14492/hokmj/1381758416 · Zbl 0409.17013 |

[17] | J. London Math. Soc 45 pp 341– (1992) |

[18] | DOI: 10.1007/BF01459487 · Zbl 0789.53021 |

[19] | Lie contact manifods pp 191– (1989) |

[20] | Lectures on differential geometry (1964) |

[21] | DOI: 10.1006/aima.1993.1002 · Zbl 0778.53041 |

[22] | Riemannian submersions, four-manifolds and Einstein-Weyl geometry 66 pp 381– (1993) · Zbl 0742.53014 |

[23] | DOI: 10.1090/S0002-9947-1970-0284936-6 |

[24] | Characteristic classes (1974) |

[25] | Twistor spaces for real four-dimensional Lorentzian manifolds 134 pp 107– (1994) · Zbl 0801.53047 |

[26] | Soviet Maht 22 pp 54– (1978) |

[27] | Gauge field theory and complex geometry (1988) |

[28] | Foundations of differential geometry I, Interscience Publishers (1963) · Zbl 0119.37502 |

[29] | On filterd Lie algebras and geometric structures I 13 pp 875– (1964) |

[30] | DOI: 10.2748/tmj/1178225150 · Zbl 0880.53052 |

[31] | Transformation groups in differential geometry (1972) · Zbl 0246.53031 |

[32] | DOI: 10.2969/jmsj/04510001 · Zbl 0790.17015 |

[33] | DOI: 10.1088/0264-9381/2/4/021 · Zbl 0575.53042 |

[34] | J. Math. Tokushima Univ 4 pp 1– (1970) |

[35] | DOI: 10.1007/BFb0066025 |

[36] | Tangent and cotangent bundles (1973) |

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