Geometric Theory of Foliations by Cesar Camacho, Alcides Lins Neto

By Cesar Camacho, Alcides Lins Neto

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Famous problems of elementary geometry: the duplication of the cube, the trisection of an angle, the quadrature of the circle: an authorized translation of F. Klein's Vorträge

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A)=M int (M A, it follows that aA is also invariant. Since aA = A - ( )). — — — — — §2. Transverse uniformity Let E be a submanifold of M. We say that E is transverse to 5 when E is transverse to every leaf of 5 that it meets. When dim ( ) + dim ( ) = = dim (M) we say E is a transverse section of Given p E M there is always a transverse section of 5 passing through p. In fact it suffices to consider a local chart ( Uop ) E with p E U, ça(U) == U, X U2 C IR" x 111 5 , ça(p) = (c,,c2 ) and take E = ça '(c 1 x U2 ).

Theorem 4. Suppose M is connected and let P be a continuous k-plane field on M. Let ( /17/, 7r, 7r* (P)) be the double covering of P. Then is orientable. ( b ) P is connected if and only if P is not orientable. (a) 7r* (P) • ( P (x)), if Proof. (a) By definition 71- * (P)(x,(e') = (D7 ( x, O )) (x, O) E /t7/. We define an orientation 0* for 7r* ( P) by saying that the ordered basis { (x, O) Xk (x,O)j of 7 * (P) (x,0) is in 0* (x) if the basis [D7r ( x, () - X' (x,0) D7r (x, O) • X k (x, O) is in O.

From now on we consider only foliations of class Cr, r 1. The charts (U, ça) E if will be called foliation charts. Let if be a cr foliation of dimension n, 0 < n < m, of a manifold Ar Consider a local chart ( U, so ) of if such that so ( U) = U1 x U2 C 112" x 1R tm ". The sets of the form so ( U1 x ci ), c E U2 are called plaques of U, or else U is so -1 / U1 x Ici : U 1 x Ici plaques of 5. Fixing c E U2, the map f a Cr imbedding, so the plaques are connected n-dimensional Cr submanifolds of M.

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