Translation foliations of codimension one on compact affine manifolds

Volume 39 / 1997

Francisco Turiel Banach Center Publications 39 (1997), 171-182 DOI: 10.4064/-39-1-171-182

Abstract

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.

Authors

  • Francisco Turiel

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