Affine bundles and integrable almost tangent structures.

*(English)*Zbl 0572.53031The authors study integrable almost tangent structures S from the affine point of view throughout the paper. They first consider in some detail: i) the concepts of fibred manifolds, bundles and fibre bundles, proving that an affine bundle modelled on a vector bundle E is a fibre bundle with structure group aff(V), the group of affine automorphisms of the standard fibre V in E; ii) almost tangent structures, first as (1,1)- tensor fields, giving three examples, and then as G-structures, considering the integrability in terms of the Nijenhuis tensor, and also the diffeomorphisms of a tangent bundle TM preserving its usual almost tangent structure.

They then consider global questions, giving as an illustration a family of integrable almost tangent structures on the torus \(T^ 2\), such that the distribution ker S\(=im S\) is tangent to the spiral which is the image of the line \(y=x \tan \alpha\) under the usual identification map \({\mathbb{R}}^ 2\to T^ 2\). Then, in order to exclude the kind of pathology which occurs in the case when \(\alpha\) is an irrational multiple of \(2\pi\), they reduce to the case of integrable almost tangent structures which define fibrations, defining such a structure S on the manifold N as one for which the quotient space by the leaves of the foliation defined by ker S is a smooth manifold M. They then define an associated vertical lift for the fibration \(\pi\) : \(N\to M\), and also study the properties of the symmetric connections adapted to S.

With these results they then prove their main theorem: Let (N,S) be an integrable almost tangent structure which defines a fibration \(\pi\) : \(N\to M\). Let \(\nabla\) be any symmetric connection on N such that \(\nabla S=0\), and suppose that with respect to the flat connection induced on it by \(\nabla\), each fibre of \(\pi\) is geodesically complete, and also that each fibre is connected and simply connected. Then N is an affine bundle modelled on TM.

They then consider global questions, giving as an illustration a family of integrable almost tangent structures on the torus \(T^ 2\), such that the distribution ker S\(=im S\) is tangent to the spiral which is the image of the line \(y=x \tan \alpha\) under the usual identification map \({\mathbb{R}}^ 2\to T^ 2\). Then, in order to exclude the kind of pathology which occurs in the case when \(\alpha\) is an irrational multiple of \(2\pi\), they reduce to the case of integrable almost tangent structures which define fibrations, defining such a structure S on the manifold N as one for which the quotient space by the leaves of the foliation defined by ker S is a smooth manifold M. They then define an associated vertical lift for the fibration \(\pi\) : \(N\to M\), and also study the properties of the symmetric connections adapted to S.

With these results they then prove their main theorem: Let (N,S) be an integrable almost tangent structure which defines a fibration \(\pi\) : \(N\to M\). Let \(\nabla\) be any symmetric connection on N such that \(\nabla S=0\), and suppose that with respect to the flat connection induced on it by \(\nabla\), each fibre of \(\pi\) is geodesically complete, and also that each fibre is connected and simply connected. Then N is an affine bundle modelled on TM.

Reviewer: P.M.Gadea

##### MSC:

53C15 | General geometric structures on manifolds (almost complex, almost product structures, etc.) |

53C12 | Foliations (differential geometric aspects) |

55R10 | Fiber bundles in algebraic topology |

##### Keywords:

almost tangent structures; affine bundle; vector bundle; fibre bundle; G- structures; integrability; Nijenhuis tensor
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\textit{M. Crampin} and \textit{G. Thompson}, Math. Proc. Camb. Philos. Soc. 98, 61--71 (1985; Zbl 0572.53031)

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