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Translation foliations of codimension one on compact affine manifolds

100%

Turiel F. J.

Banach Center Publications

1997

tom 39

nr 1

171-182

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

Classification of (1,1) tensor fields and bihamiltonian structures

100%

Turiel F. J.

Banach Center Publications

1996

tom 33

nr 1

449-458

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 $(df_{1} ∧ ... ∧ df_{m})(p) ≠ 0$ and $d(df_{j}(J( )))(p) = 0$, 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.

Ograniczanie wyników

2 Banach Center Publications

2 Turiel F. J.

1 1997

1 1996

Wyniki wyszukiwania

Translation foliations of codimension one on compact affine manifolds

Classification of (1,1) tensor fields and bihamiltonian structures