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Consider two foliations ${ℱ}_1$ and ${ℱ}_2$, of dimension one and codimension one respectively, on a compact connected affine manifold $(M,\nabla )$. Suppose that ${\nabla }_{T{ℱ}_1}T{ℱ}_2\subset T{ℱ}_2$; ${\nabla }_{T{ℱ}_2}T{ℱ}_1\subset T{ℱ}_1$ and $TM=T{ℱ}_1\oplus T{ℱ}_2$. In this paper we show that either ${ℱ}_2$ is given by a fibration over $S^1$, and then ${ℱ}_1$ has a great degree of freedom, or the trace of ${ℱ}_1$ is given by a few number of types of curves which are completely described. Moreover we prove that ${ℱ}_2$ has a transverse affine structure.

Consider a (1,1) tensor field J, defined on a real or complex m-dimensional manifold M, whose Nijenhuis torsion vanishes. Suppose that for each point p ∈ M there exist functions $f_1,...,f_m$, defined around p, such that $(d{f}_1\wedge ...\wedge d{f}_m)\left(p\right)\ne 0$ and $d\left(d{f}_j\left(J\left(\right)\right)\right)\left(p\right)=0$, j = 1,...,m. Then there exists a dense open set such that we can find coordinates, around each of its points, on which J is written with affine coefficients. This result is obtained by associating to J a bihamiltonian structure on T*M.

An elementary proof of the following theorem is given: THEOREM. Let M be a compact connected surface without boundary. Consider a C^∞ action of Rⁿ on M. Then, if the Euler-Poincaré characteristic of M is non zero there exists a fixed point.