Showing papers on "Distribution (differential geometry) published in 1978"

The order and symbol of a distribution

Abstract: A definition is given, for an arbitrary distribution g on a manifold X, of the order and symbol of g at a point (x, ?) of the cotangent bundle TX. If X = R', the order of g at (0, $) is the growth order as X co of the distributions gT(x) = e < g(x/ ); if the order is less than or equal to N, the N-symbol of g is the family gt modulo O (T N 1/2). It is shown that the order and symbol behave in a simple way when g is acted upon by a pseudo-differential operator. If g is a Fourier integral distribution, suitable identifications can be made so that the symbol defined here agrees with the bundle-valued symbol defined by Hormander. PREFACE Since the introduction of pseudo-differential operators, the analysis of distributions on a manifold X has involved the geometry of the cotangent bundle T*X. With the notion of wavefront set [6], one can localize a distribution at a nonzero cotangent vector ( to obtain its "microgerm", just as one localizes at x E X to obtain the ordinary germ. On the base space, X, one can go beyond the local level to the infinitesimal one; for a C function, the result is its "jet", which can be thought of as a function on the tangent space T,X. In this paper, we describe an analogous procedure for distributions: given a distribution g on X and a cotangent vector {, we construct a jet-like object called its symbol, which is a distribution on TX depending on certain parameters. (In [13], we show that the symbol may be thought of as an object on TCT*X, thus completing the analogy with jets.) In Chapter I, we define the symbol and the space in which it lies, we prove the elementary properties of the symbol, and we give some examples. Chapter II, written in collaboration with K. Sklower, establishes the relation between our symbol and the wavefront set. In Chapter III, we show that our symbol construction contains the one given by Hormander [6] for a very special class of distributions-the Fourier integral distributions. This last result was the basis for the whole paper. In lectures at the Nordic Received by the editors November 24, 1976. AMS (MOS) subject classifications (1970). Primary 46F10; Secondary 53C15, 58G15. C American Mathematical Society 1978

Nullity and generalized characteristic classes of differential manifolds

Abstract: Using the Kamber-Tondeur construction of characteristic classes for foliated bundes, the author has given a method for constructing generalized characteristic classes for a differentiable manifold M without imposing conditions on M. In particular a vanishing theorem on the manifold M is obtained. The construction is particularly useful if the ordinary characteristic ring Pont*(M) of the manifold M vanishes much below the dimension of M. 1. Introduction. The theory of generalized (ordinary and secondary) characteristic classes of a manifold M which is associated with a certain geometric structure is a recent achievement in differential geometry. In contrast to the ordinary characteristic classes, the newly constructed secondary classes carry with them underlying geometric structures. The construction of secondary classes began in a series of papers by Bott (1), Bott and Haefliger (2), Chern and Simons (5), Godbillon and Vey (6), Kamber and Tondeur (7), (8) and others. The idea of the construction is as follows, namely, if the characteristic form f(K) is zero, then the transgression Tf(K) of f(K) defines a secondary class in the principal fibre bundle (5); here / is an invariant polynomial and K is the curvature form. Kamber and Tondeur found the usefulness of the Weil algebra W{G) of a Lie group G in this context. They showed that for a foliated bundle (P, E, {«}) (7), Bott's vanishing theorem (1), when interpreted in the Weil homomorphism (3), k(w): W(G)->A(P), gave rise to a new generalized characteristic homomorphism A,: H(W(G, H)q) -» H(M, R) where E denotes a foliation on the manifold M of codimension q, {co} is the family of adapted connections and W(G, H)q denotes the truncated relative Weil algebra (7), (8). We are able to define generalized characteristic classes on an affine (or Riemannian) manifold M using the Cartan-Kamber-Tondeur map At. We can do this by observing that the affine {or Riemannian) manifold is already foliated by its nullity distribution (§2). This construction is particularly useful if