Complementary distributions which preserve the leaf geometry and applications to totally geodesic foliations.

*(English)*Zbl 0572.57016Let \(F\) 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 [TĂ´hoku 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.}

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 [TĂ´hoku 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) |