Showing papers on "Frame bundle published in 1991"

Projective manifolds whose tangent bundles are numerically effective.

Abstract: In 1979, Mori [Mo] proved the so-called Hartshorne-Frankel conjecture: Every projective n-dimensional manifold with ample tangent bundle is isomorphic to the complex projective space P,. A differential-geometric analogon assuming the existence of a K/ihler metric on X with positive holomorphic bisectional curvature is independently due to Siu-Yau [SY]. Thus it seems natural to classify projective manifolds X whose tangent bundle Tx satisfy a degenerate condition of ampleness: numerical effectivity (abbreviated by "nef'). This means that the tautological quotient line bundle d~(1) on F(Tx) is numerically effective, i.e. C_>_0

Structure of spacetime tangent bundle

Abstract: The universal upper limit on attainable proper acceleration relative to the vacuum imposes restrictions on possible structures in the spacetime tangent bundle. Various features of the differential geometry of the spacetime tangent bundle are presented here. Also, a modified Schwarzschild solution is obtained, and the associated gravitational red shift is calculated.

String structures and the path fibration of a group

Abstract: We use the realisation of the universal bundle for the loop group as the path fibration of the group to investigate the string class, that is the obstruction to a loop group bundle lifting to a Kac-Moody group bundle. In the case that the loop group bundle is constructed by taking loops into a principal bundle we show that the classifying map is the holonomy around loops and give an explicit formula for the string class relating it to the Pontrjangin class of the principal bunble.

Lagrangians for Ricci-flat geometries

Abstract: The new nonchiral Lagrangian density for the Einstein field equations, which leads to Nester's frames and Hamiltonian formulations in Ashtekar variables, is shown to arise naturally as a canonical 4-form on the second frame bundle, of pure grade 4 in m=4 Clifford algebra. This insight allows such Lagrangians to be written as easily for higher dimensional theories.

On the 2 very ampleness of the adjoint bundle

Abstract: Let S be a smooth projective surface over C polarized by a 2-very ample line bundle L=O S(L), i.e. for any 0-dimensional subscheme (Z,O Z ) of length 3 the restriction map Γ(L)→Γ(L⊗O Z) is a surjection. This generalization of very ampleness was recently introduced by M. Beltrametti and A.J. Sommese. The authors prove that, if L·L≥13, the adjoint line bundleK S⊗L is 2-very ample apart from a list of well understood exceptions and up to contracting down the smooth rational curves E such that E·E=−1, L·E=2. The appendix contains an inductive argument in order to extend the result in higher dimension.

Gauge transformations and fiber bundle theory

Abstract: The relationship between the group Γ of pure gauge transformations of Atiyah, Hitchin, and Singer [Proc. R. Soc. London Ser. A 362, 425 (1978)] of a principal fiber bundle and the group G of gauge transformations consisting of the direct product of the local gauge groups on the base space is studied. Γ is an invariant subgroup of G and the quotient G/Γ is identified with the group of inequivalent gauge transformations. In the framework of the category of principal fiber bundles with connections, a natural explanation for the relevance of the group Γ in the classical and quantum theories of gauge fields is presented. The paper is made self‐contained by an introductory discussion of the concepts of principal coordinate fiber bundle (gauge fixed principal fiber bundle) and principal fiber bundle, and of the equivalence between the three different versions of the group of vertical automorphisms of the bundle.

On the characteristic classes of actions of lattices in higher rank Lie groups

Abstract: We show that under certain assumptions, the measurable cohomology class of the linear holonomy cocycle of a foliation yields information about the characteristic classes of the foliation. Combined with the results of a previous paper, this yields vanishing theorems for characteristic classes of certain actions of lattices in higher rank semisimple Lie groups. Let F be a discrete group acting by diffeomorphisms on a smooth compact manifold M. Associated to this action are certain characteristic classes in H (F, R), which are constructed as characteristic classes of a natural foliation associated to the group action. The action of F on M induces an action of F on the principal frame bundle P(M) of M and the characteristic classes of the action can be interpreted as obstructions to the existence of invariant geometric structures on M, i.e., principal subbundles of P(M) invariant by the F-action. If F is a lattice in a higher rank semisimple Lie group, then F has strong rigidity properties (see, e.g., [M, Z1]). In an earlier paper [S], we showed, using techniques from ergodic theory, that for a certain class of F-actions there is always an invariant measurable reductive geometric structure, i.e., a measurable principal subbundle with reductive structure group, which is invariant by the F-action. Moreover, the noncompact semisimple part of this reductive group is locally isomorphic to a semisimple factor of the ambient Lie group of F [Zi]. Zimmer [Z3] recently proved this result for a large class of actions (which does not a priori include the class of actions considered in [S]). A natural question is whether these results remain true in the smooth category. The purpose of this paper is to show that the characteristic classes, which obstruct a smooth geometric structure, vanish in the presence of a measurable geometric structure. Explicitly, we have Main Theorem. Let (M, Y) be a codimension n, C 2-foliated manifold and suppose that the linear holonomy cocycle is measurably equivalent to a locally tempered cocycle taking values in a subgroup H c GL(n, R) which is stable under transpose. Then the Weil homomorphism X: H'(g[(n), 0(n)) -Hc(M, Yi) Received by the editors February 24, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 57R32; Secondary 57S20. Research partially supported by an Alfred P. Sloan Dissertation Fellowship. ? 1991 American Mathematical Society 0002-9947/91 $1.00 + $.25 per page

Geometry of Low-dimensional Manifolds: Extremal immersions and the extended frame bundle

Abstract: We present a computationally powerful formulation of variational problems that depend on the extrinsic and intrinsic geometry of immersions into a manifold. The approach is based on a lift of the action integral to a larger space and proceeds by systematically constraining the variations to preserve the foliation of a Pfaffian system on an extended frame bundle. Explicit Euler-Lagrange equations are computed for a very general class of Lagrangians and the method illustrated with examples relevant to recent developments in theoretical physics. The method provides a means of determining spatial boundary conditions for immersions with boundary and enables a construction to be made of constants of the motion in terms of Euler- Lagrange solutions and admissible symmetry vectors.

Geometric Quantization on Wiener Manifolds

Abstract: The Geometric Quantization procedure is considered in the context of Wiener manifolds. Introduction. The material presented here may be viewed as a continuation of the discussion in (1) where some basic aspects of the problem of extending the geometric quantization procedure to the infinite dimensional case were discussed. In (3)(24)(25) the problem of quantizing a linear phase space has been considered by taking the limit of results obtained in finite dimensional geometric quantization. Many of these objects, e.g. the trace of certain linear mappings, the Laplace op­ erator and Kiihler potentials, do not have a well defined limit as n goes to infinity and have to be taken care of by hand. Here we construct a geometric quantization­ procedure on the infinite dimensional phase space and obtain Bargmann spaces, the Bogoliubov transformation and infinitesimal pairing expressed in terms of well defined objects on a Wiener space. Let H be a (strongly) symplectic Hilbert space with a complex structure. Given a Lagrangian subspace Wo in He, the restricted Siegel upper half plane K+ is defined as the set of AWo for A E BpT! the restricted symplectic group. The vacuum in the Fock space B'(W) defined w.r.t. W defines a line bundle 8 over K+. This line bundle plays the role in our work of the bundle of half forms which occurrs in the BKS pairing in the finite dimensional case. As discussed in (16) a U(l) extension MPr of Spr acts on c. The arena for GQ in the infinite dimensional case considered here is Wiener manifolds. We use a technical setup based on the work of Kusuoka (10). This may be expected to generalize further to a setup based entirely on the concept of smoothness defined in terms of the Frechet spaces Woo used in the Malliavin calculus. In contrast to ego the definition of Wiener manifold used in the work of Piech (13) and Eells-Elworthy (7) the definition given below does not imply that a Wiener manifold M is a Banach manifold. Rather we consider regularity conditions defined purely in terms of measurability and regularity in H-directions. In this setting, a Wiener manifold has a natural tangent bundle with Hilbert space structure on the fibers (but no tangent bundle with fibers modelled on B) and a Gi r reduction of the frame bundle. Supported in part by NFR, the Swedish Academy of Sciences and the Gustavsson foundation Typeset by AMS-'lEX

Algebraic hulls and smooth orbit equivalence

Abstract: Let (Mi,,ui), i = 1, 2, be two manifolds with quasi-invariant measures, and let Hi c Diff(Mi) be connected Lie groups. If there is a measure class preserving diffeomorphism 0: M1 - > M2 which is a bijection of Hl-orbits and H2-orbits then we say that the actions are smoothly orbit equivalent. If the Hi-actions determine foliations g on the manifolds MiS then the map 0 is just a diffeomorphism between the foliations. These phenomena, and the corresponding phenomena arising in situations in which the (Mi, Ai) are just Borel (or topological) Hi-spaces and the map 0 is a measure class preserving Borel isomorphism (or homeomorphism), have been studied, using a variety of techniques, independently by several authors [B, CFW, D1, D2, K, PnZ, W1, W2, Z2, Z4]. Most of the results obtained require some strengthening of the hypotheses, such as finiteness and invariance of the measures and amenability or semisimplicity (in higher rank) of the groups acting. One of the results that we want to describe in this paper fits in this geometric setting and provides an obstruction, in terms of a geometric invariant of the actions, to the foliations being diffeomorphic. Recall that, if H acts ergodically and by diffeomorphisms on the n-dimensional manifold M, the algebraic hull of the H-action is the unique (up to conjugacy) smallest algebraic subgroup L c GL(n, R) such that there exists a measurable H-invariant reduction to L of the frame bundle on M on which H acts by automorphisms. (For an analytic definition see §2 and for the general context see [Z5, 9.2].) Then the normal algebraic hull of the H-action on M will be the projection of L in the direction normal to the orbits.

The diffeomorphism group and flat principal bundles

Abstract: Representations of the semidirect product group D■Diff(M) in the context of fiber bundle theory are studied. Here, D is the group of compactly supported functions and Diff(M) is the group of compactly supported diffeomorphisms of a manifold M. The carrier space is taken as the space of equivariant functions on a flat principal bundle over M where M is multiply connected. Two principal bundles are taken as equivalent if they are related by a gauge transformation. For the U(1) case it is found that the representations are irreducible, and the equivalence classes of representations are in one to one correspondence with the equivalence classes of bundles. Simple examples are discussed.