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Affine bundles and integrable almost tangent structures. (English) Zbl 0572.53031
The 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.