Showing papers on "Frame bundle published in 1974"

Transversally parallelizable foliations of codimension two

Abstract: We study framed foliations such that the framing of the normal bundle can be chosen to be invariant under the linear holonomy of each leaf. In codimension one there is a strong structure theory for such foliations due, e.g., to Novikov, Sacksteder, Rosenberg, Moussu. An analogous theory is developed here for the case of codimension two. Introduction. Let M be a smooth (i.e., C') connected manifold, '5 a smooth foliation of M of codimension q, Q the normal bundle of C, and F(Q) the normal frame bundle. If H C Glq is a Lie subgroup, a transverse H-structure for T will be a smooth reduction of F(Q) to an H-bundle which is invariant under the natural parallelism along the leaves of i. Such an H-reduction will be said to be "compatible" with 5 (more precise definitions appear in ? 1). A foliation together with a choice of transverse H-structure will be called an H-foliation. In this way, for instance, the foliated manifolds with bundle-like metric of Reinhart [14] are interpreted simply as Oq-foliations and the Riemannian foliations of Pasternack [13] as foliations admitting a transverse Oq-structure, while transversally orientable foliations are those with a transverse Glq-structure. It seems reasonable to investigate the topological and geometric consequences of the existence of transverse H-structures for the various Lie groups H. In this paper we carry out such investigations for the extreme cases in which H is discrete (transversally almost parallelizable foliations) and those in which H = e is trivial (transversally parallelizable or e-foliations). As examples, the standard foliations of the torus Tn induced by parallel (n q)-planes in R' are transversally parallelizable, while the foliation of an open Mobius strip by the curves parallel to the center circle is transversally almost parallelizable. We have a number of general results for such foliations valid in arbitrary codimension (cf. ??3 and 4), but our strongest theorems are for codimension two and require M to be compact. In codimension one, e-foliations of compact manifolds are rather well understood. In this case, our condition is equivalent to the absence of limit cycles, a situation which has been studied by a variety of authors (e.g., [9], [11], [12]). In particular, a structure theorem due to Novikov [12] asserts that the universal cover M of M will have the form A x R where A is the universal cover of the Received by the editors May 3, 1973. AMS (MOS) subject class'iocations (1970). Primary 57D30. (') Research partially supported by NSF Grant GP-20842. Copyright K 1974, American Mathematical Society

Foliations transverse to fibers of a bundle

Abstract: Consider a fiber bundle where the base space and total space are compact, connected, oriented smooth manifolds and the projection map is smooth. It is shown that if the fiber is null-homologous in the total space, then the existence of a foliation of the total space which is transverse to each fiber and such that each leaf has the same dimension as the base implies that the fundamental group of the base space has exponential growth. Introduction. Let (E, p, B) be a locally trivial fiber bundle where /»:£-*-/? is the projection, E and B are compact, connected, oriented smooth manifolds and p is a smooth map. (By smooth we mean C for some r^l and, henceforth, all maps are assumed smooth.) Let b denote the dimension of B and k denote the dimension of the fiber (thus, dim E= b+k). By a section of the fibration we mean a smooth map a:B-+E such that p o a=idB. It is well known that if a section exists then the fiber over any point in B represents a nontrivial element in Hk(E; Z) since the image of a section is a compact orientable manifold which has intersection number one with the fiber over any point in B. The notion of a section may be generalized as follows. Definition. A polysection of (E,p,B) is a covering projection tr:B-*B together with a map C:B-*E such that the following diagram commutes. Note that if the covering projection is a diffeomorphism then the map f o 7T_1 is a section in the usual sense. One important situation in which polysections arise is the following. Suppose that F is a smooth foliation of E such that each leaf of F is Received by the editors April 4, 1973 and, in revised form, May 1, 1973. AMS (MOS) subject classifications (1970). Primary 57D30; Secondary 55F10.

Euler Classes of Stably Equivalent Vector Bundles

Abstract: The possible values of the Euler class of a k-dimensional bundle stably equivalent to a fixed k-dimensional real vector bundle are studied. Complete information is obtained in the case where the dimension of the bundle is more than half of the dimension of the base. In the case where k is an arbitrary even number it is shown that if the original bundle possesses a vector field, then for every element of the corresponding cohomology group whose square is equal to zero there exists an element divisible by it realized as the Euler class of a bundle stably equivalent to the original.Bibliography: 13 items.

linear vector fields on manifolds

Abstract: The classical study of a flow near a fixed point is generalized by composing, at each point in the manifold, the flow derivative with a parallel translation back along the flow. Circumstances under which these compositions form a one-parameter group are studied. From the point of view of the linear frame bundle, the condition is that the canonical lift commute with its horizontal part (with respect to some metric connection). The connection form applied to the lift coincides with the infinitesimal generator of the one-parameter group. Analysis of this matrix provides dynamical information about the flow. For example, if such flows are equicontinuous, they have uniformly bounded derivatives and therefore the enveloping semigroup is a Lie transformation group. Subclasses of ergodic, minimal, and weakly mixing flows with integral invariants are determined according to the eigenvalues of the matrices. Such examples as Lie algebra flows, infinitesimal affine transformations, and the geodesic flows on manifolds of constant negative curvature are examined. 0. Introduction. This work grew out of an attempt to generalize the classical consideration of a vector field on a manifold through analysis of the flow derivatives. Classically, if IXt It reall is the flow of X with fixed point p, then tdXt(p) J t reall forns a one-parameter group of transformations of T (M). The properties of the flow are determined near p by the infinitesimal generator of this group. If one drops the assumption that p is a fixed point, the problem of identifying Tp(M) with TXW(p)(M) arises. One possibility is to provide M with a Riemannian structure and to identify Txt(p)(M) and Tp(M) via parallel translation along the flow. Unfortunately, the resulting set of transformations of T. (M) need not form a group. Conditions under which these transformations form a group are investigated. The entire situation becomes clearer from the point of view of the linear frame bundle, L(M). If X is the natural lift of X to L(M) and c is a metric Presented to the Society, November 3, 1973; received by the editors February 21, 1973. AMS (MOS) subject classifications (1970). Primary 34C35, 54H20, 58F99; Secondary 28A65, 57E15.

Quasilinear resolutions of non-linear equations

Abstract: By analogy with the linear vector bundle case, a non-linear partial differential equation on a manifold can be defined as a fibred submanifold Rk of a k-jet bundle. By observing that under natural conditions the first prolongation gives rise to a vector bundle over Rk, (that is, a quasilinear equation), techniques of the linear case are adapted to establish conditions for the formal integrability of the equation.