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Complementary distributions which preserve the leaf geometry and applications to totally geodesic foliations. (English) Zbl 0572.57016
Let $$F$$ be a smooth foliation on a manifold $$M$$. Then a complementary distribution $$D$$ is called an Ehresmann connection if every two paths $$\sigma$$ of $$D$$ (i.e., tangent to $$D$$), $$\tau$$ of $$F$$ starting at the same point $$p\in M$$ are the edges of a rectangle of paths of $$D$$ and $$F$$. Such a $$D$$ allows for thedefinition of a quotient group $$H_ D(L,p)=\Pi_ 1(L,p)/K_ D(L,p)$$, where $$L$$ is a leaf of $$F$$, and $$K_ D(L,p)$$ is defined by ”$$D$$-preserving loops”. The authors prove that, if a leaf $$L_ 0$$ is compact, and $$H_ D(L_ 0,p_ 0)$$ is finite, the same holds for every leaf $$L$$. If $$M$$ is a complete Riemannian manifold, if F is a totally geodesic foliation, and if a leaf $$L_ 0$$ has $$H_ D(L_ 0,p_ 0)$$ finite, and finite volume, the same holds for every leaf $$L$$.
Finally, in both the general and the totally geodesic case, if $$D$$ preserves a complete parallelism of the leaves, one describes the geometric structure of $$M$$ by means of the submanifolds $$P(x)$$ defined by points which can be joined to $$x$$ ($$x\in M$$) by paths of $$D$$.
{Remark. Integrable Ehresmann connections have been studied as ”latticed maps” by S. Kashiwabara [Tôhoku Math. J., II. Ser. 11, 43-53 (1959; Zbl 0131.199)]. The problem with the Ehresmann connections is that, except for the totally geodesic complete case, there are no good means to decide whether such connections exist.}
Reviewer: I.Vaisman

##### MSC:
 57R30 Foliations in differential topology; geometric theory 53C12 Foliations (differential geometric aspects)
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