Showing papers on "Frame bundle published in 1988"

Differential Geometry of Frame Bundles

Abstract: 1 The Functor Jp1.- 1.1 The Bundle Jp1M ? M.- Functorial properties of Jp1.- Canonical lifts of vector fields to Jp1M.- Two particular cases.- Diffeomorphisms ? Mp,1 and ? M1,p.- 1.2 Jp1G for a Lie group G.- Jp1G acting on Jp1M.- 1.3 Jp1V for a vector space V.- Jp1g for a Lie algebra g.- 1.4 The embedding jp.- V = Rn.- 2 Prolongation of G-structures.- 2.1 Imbedding of Jn1FM into FFM.- 2.2 Prolongation of G-structures to FM.- 2.3 Integrability.- 2.4 Applications.- Linear endomorphisms.- Bilinear forms.- Linear groups.- 3 Vector-valued differential forms.- 3.1 General Theory.- Particular cases.- V =? s1Rn.- V =? sRn.- 3.2 Applications.- Prolongation of functions and forms.- Complete lift of functions and tensor fields.- Prolongation of G-structures.- 4 Prolongation of linear connections.- 4.1 Forms with values in a Lie algebra.- 4.2 Prolongation of connections.- Local expressions.- Covariant differentiation operators.- 4.3 Complete lift of linear connections.- Parallelism.- 4.4 Connections adapted to G-structures.- 4.5 Geodesics of ?C.- 4.6 Complete lift of derivations.- 5 Diagonal lifts.- 5.1 Diagonal lifts.- 5.2 Applications.- G-structures from (1, 1)-tensors.- G-structures from (0, 2)-tensors.- General tensor fields.- 6 Horizontal lifts.- 6.1 General theory.- 6.2 Applications.- Tensor fields.- Linear connections.- Geodesics.- Covariant derivative.- Canonical flat connection on FM.- Derivations.- 7 Lift GD of a Riemannian G to FM.- 7.1 GD, G of type (0,2).- 7.2 Levi-Civita connection of GD.- 7.3 Curvature of GD.- 7.4 Bundle of orthonormal frames.- 7.5 Geodesics of GD.- 7.6 Applications.- f-structures on FM.- Almost Hermitian structure.- Harmonic frame bundle maps.- 8 Constructing G-structures on FM.- 8.1 ?-associated G-structures on FM.- 8.2 Defined by (1,1)-tensor fields.- 8.3 Application to polynomial structures on FM.- Example 1: f(3, 1)-structure on FM.- Example 2: f(3, -1)-structure on FM.- Example 3: f(4,2)-structure on FM.- Example 4: f(4, -2)-structure on FM.- Example 5: A family of examples.- 8.4 G-structures defined by (0,2)-tensor fields.- 8.5 Applications to almost complex and Hermitian structures.- 8.6 Application to spacetime structure.- 9 Systems of connections.- 9.1 Connections on a fibred manifold.- Local expressions.- Examples of linear connections.- Notation for sections.- 9.2 Principal bundle connections.- Summary for the principal bundle of frames.- 9.3 Systems of connections.- Examples of systems of linear connections.- 9.4 Universal Connections.- 9.5 Applications.- Universal holonomy.- Weil's Theorem.- Spacetime singularities.- Parametric models in statistical theory.- 10 The Functor Jp2.- 10.1 The Bundle Jp2M ? M.- Functorial properties of Jp2.- 10.2 The second order frame bundle.- 10.3 Second order connections.- 10.4 Geodesics of second order.- 10.5 G-structures on F2M.- 10.6 Vector fields on F2M.- 10.7 Diagonal lifts of tensor fields.- Algebraic preliminaries.- Diagonal lifts of 1-forms.- Diagonal lifts of (1, 1)-tensor fields.- Diagonal lifts of (0, 2)-tensor fields.- F2M for an almost Hermitian manifold M.- 10.8 Natural prolongations of G-structures.- Imbedding of Jn2FM into FF2M.- Applications.- Linear endomorphisms.- Bilinear forms.- 10.9 Diagonal prolongation of G-structures.- Applications.- Linear endomorphisms.- Bilinear forms.

Vanishing theorems for tensor powers of an ample vector bundles.

Abstract: LetE be a holomorphic vector bundle of rankr on a compact complex manifoldX of dimensionn. It is shown that the cohomology groupsH p,q (X, E⊗k ⊗(detE) l ) vanish ifE is ample andp+q≧n+1,l≧n−p+r−1. The proof rests on the well-known fact that every tensor powerE ⊗k , splits into irreducible representations of Gl(E). By Bott's theory, each component is canonically isomorphic to the direct image onX of a homogeneous line bundle over a flag manifold ofE. The proof is then reduced to the Kodaira-Akizuki-Nakano vanishing theorem for line bundles by means of the Leray spectral sequence, using backward induction onp. We also obtain a generalization of Le Potier's isomorphism theorem and a counterexample to a vanishing conjecture of Sommese.

Triad Approach to the Hamiltonian of General Relativity

Abstract: The Sparling-Thirring forms are constructed using a connection over a frame bundle defined in terms of a spacelike triad and lapse and shift functions. From the forms, one can identify a first-order Lagrangian from which to construct the Hamiltonian using triad vector densities as basic configuration-space variables. If one uses the self-dual part of the forms, the resulting first-order Lagrangian (its imaginary part is a total divergence) leads to momenta which are the dual of the Sen connection which was introduced by Ashtekar in a spinor representation. The resulting formalism is equivalent to that of Ashtekar.

Characteristic classes of surface bundles and bounded cohomology

Abstract: Publisher Summary This chapter discusses characteristic classes of surface bundles and bounded cohomology. The isomorphism class of a surface bundle is completely determined by the induced homomorphism. If Σg is a closed orientable surface of genus g, then a differentiable fibre bundle π:E→X, whose fibre is diffeomorphic to Σg, is called a surface bundle or a Σg- bundle. The characteristic classes of surface bundles are highly nontrivial. The chapter presents a sufficient condition for the characteristic classes of surface bundles to vanish. An abstract group is called amenable, if there exists a left invariant mean on the set of all bounded real valued.

Gauge theory of a group of diffeomorphisms. III. The fiber bundle description

Abstract: A new fiber bundle approach to the gauge theory of a group G that involves space‐time symmetries as well as internal symmetries is presented. The ungauged group G is regarded as the group of left translations on a fiber bundle G(G/H,H), where H is a closed subgroup and G/H is space‐time. The Yang–Mills potential is the pullback of the Maurer–Cartan form and the Yang–Mills fields are zero. More general diffeomorphisms on the bundle space are then identified as the appropriate gauged generalizations of the left translations, and the Yang–Mills potential is identified as the pullback of the dual of a certain kind of vielbein on the group manifold. The Yang–Mills fields include a torsion on space‐time.

Automorphic vector bundles on connected Shimura varieties

Abstract: Introduction 0. Review of terminology concerning Shimura varieties 1. The Taniyama group, the period torsor, and conjugates of Shimura varieties 2. The compact dual symmetric Hermitian space and its conjugates 3. The principal bundle Y~ X); statement of the first main theorem 4. Automorphic vector bundles 5. Conjugates of automorphic vector bundles 6. Proof of Theorem 3.10 for the symplectic group 7. Proof of Theorem 3.10 for connected Shimura varieties of abelian type 8. First completion of the proof of Theorem 3.10 9. Second completion of the proof of Theorem 3.10 Appendix: Pairs defining connected and nonconnected Shimura varieties Bibliography 91 93 95 101 105 112 115 118 121 123 124 126 128

The restricted tangent bundle of a rational curve in P2

Abstract: (1988). The restricted tangent bundle of a rational curve in P2. Communications in Algebra: Vol. 16, No. 11, pp. 2193-2208.

Vanishing theorems for tensor powers of a positive vector bundle

Abstract: Let E be a holmorphic vector bundle of rank r over a compact complex manifold X of dimension n It is shown that the Dolbeault cohomology groups H p,q (X, E ⊗k ⊗(det E)l) vanish if E is positive in the sense of Griffiths and p+q≥n+1, l≥r+C (n, p, q) The proof rests on the wellknown fact that every tensor power E ⊗k splits into irreducible representations of Gl(E), each component being canonically isomorphic to the direct image on X of a positive homogeneous line bundle over a flag manifold of E The vanishing property is then obtained by a suitable generalization of Le Potier's isomorphism theorem, combined with a new curvature estimate for the bundle of X-relative differential forms on the flag manifold of E

The space of cross sections of a bundle

Abstract: Let B be a nondiscrete compactum, Y a separable complete metrizable ANR with no isolated point and p: X -B a locally trivial bundle with fiber Y admitting a section. It is proved that the space r(X) of all cross sections of p: X -B is an 12-manifold. 0. Introduction. Through the paper, spaces are separable metrizable and maps are continuous. Let p: X -* B be a locally trivial bundle with fiber Y, that is, each point b E B has a neighborhood U and a homeomorphism p: U x Y -, p-1 (U) such that p'p = ru, the projection to U. A map s: B -* X is called a cross section of p: X -* B provided ps = id. The space of all cross sections of p: X -* B with compact-open topology is denoted by 17(X). Then 17(X) is a closed subspace of the space C(B, X) of all maps from B into X. If B is compact and d is a compatible metric for X, the topology of 17(X) (and C(B, X)) is induced by the sup-metric d(f, g) = sup{d(f (b), g(b)) I b E B}. A manifold modeled on Hilbert space 12 is called an 12-manifold. In this note, we prove the following MAIN THEOREM. Let B be a nondiscrete compactum, Y a complete metrizable ANR with no isolated point and p: X B a locally trivial bundle with fiber Y admitting a section. Then 17(X) is an 12-manifold. For the trivial bundle 7rB: B x Y -* B, the space F(B x Y) can be regarded as the space C(B, Y). Thus the space C(B, Y) is an 12-manifold if B is a nondiscrete compactum and Y is a complete-metrizable ANR with no isolated point. This is a generalization of Eells-Geoghegan-Torunczyk's result [E, Ge, To1]. The author would like to thank Doug Curtis for helpful comments. 1. Preliminaries. Our proof is based on the following: TORU?NCZYK'S CHARACTERIZATION THEOREM FOR 12-MANIFOLDS [TO2] (CF. [To3]). A complete-metrizable ANR X is an 12-manifold if and only if X has the discrete approximation property, that is, for each map f: eDnEN I' X of the free union of n-cells (n E N) into X and each map E: X -+ (0, 1) there is a Received by the editors February 27, 1987 and, in revised form, April 22, 1987. Presented to the Mathematical Society of Japan, April 2, 1988. 1980 Mathematics Subject Classification. Primary 58D15, 57N20, 55F10.

Character-valued index theorems in supersymmetric string theories

Abstract: By using (1+1)-dimensional supersymmetric sigma models, the local expressions for various G*U(1) character-valued topological indices in closed supersymmetric string theories are constructed explicitly, where U(1) is the group of constant rotations along the closed string and G is the non-trivial automorphism group of the vector bundle or the tangent bundle over which the string sigma model is defined.

Controlled homotopy topological structures

Abstract: Let p : E —> B be a locally trivial fiber bundle between closed manifolds where dim E > 5 and B has a handlebody decomposition. A controlled homotopy topological structure (or a controlled structure^ for short) is a map /: M —> E where M is a closed manifold of the same dimension as E and / is a p~ι (e)-equivalence for every e > 0 (see §2). It is the purpose of this paper to develop an obstruction theory which answers the question: when is f homotopic to a homeomorphism, with arbitrarily small metric control measured in B? This theory originated with an idea of W. C. Hsiang that a controlled structure gives rise to a cross-section of a certain bundle over B, associated to the Whitney sum of p : E —• B and the tangent bundle of B.

The Extended Phase Space of the {BRS} Approach

Abstract: The origin of the classical BRS symmetry is discussed for the case of a first class constrained system consisting of a 2n-dimensional phase spaceS with free action of a Lie gauge groupG of dimensionm. The extended phase spaceSext of the Fradkin-Vilkovisky approach is identified with a globally trivial vector bundle overS with fibreL*(G)⊕L(G), whereL(G) is the Lie algebra ofG andL*(G) its dual. It is shown that the structure group of the frame bundle of the supermanifoldSext is the orthosymplectic group OSp(m,m; 2n), which is the natural invariance group of the super Poisson bracket structure on the function spaceC∞(Sext). The action of the BRS operator ω is analyzed for the caseS=R2n with constraints given by pure momenta. The breaking of the osp(m,m; 2n)-invariance down to sp(2n−2m) occurs via an intermediate “osp(m; 2n−m).” Starting from a (2n+2m)-dimensional system with orthosymplectic invariance, different choices for the BRS operator correspond to choosing different 2n-dimensional constraint supermanifolds inSext, which in general characterize different constrained systems. There is a whole family of physically equivalent BRS operators which can be used to describe a particular constrained system.

On 2-spannedness for the adjunction mapping

Abstract: LetL be a line bundle on a smooth connected projective manifold X of dimension n. We extend to any dimension the definition of k-spannedness forL; this is a notion of “k-th order embedding” which was recently given in the case of curves and surfaces. Then, by a reduction to the surfaces case, we prove that the adjoint bundle Kx+(n−1)L is 2-spanned ifL is (at least) 3-spanned.

Computing principal direction fields as frame bundle cross sections

Abstract: Essential information about the differential structure of smooth surfaces is embodied in their principal curvature and principal direction fields. A particular type of fibre bundle, a frame bundle, is defined over estimated surface trace points and the computation of these fields is presented as the determination of the appropriate cross section through the frame bundle. Any approach to the recovery of surface structure must deal with the inherent limitations of local computations-an iterative constraint-satisfaction process is developed for the reliable extraction of the principal-direction cross section. Present methods are applied to the synthetic images corrupted by noise and to magnetic resonance imagery. >

Some global properties and invariance of bundle metrics in the Kaluza–Klein scheme

Abstract: It is shown that the transition functions that give the global structure of the fiber bundle play an important role in the construction of the metric. The invariance properties of this metric under general gauge transformations are discussed and it is found that the usual requirement of a gauge‐invariant metric leads to severe constraints on the gauge fields. To avoid them, it is shown that the metric should instead be covariant with respect to these transformations. Moreover the existence of global actions that are essential in the context of the consistency problem is also discussed. The presence of such actions is studied in both the principal and their associated bundles. In the case of a homogeneous bundle with G/H as the typical fiber, it is shown that a ‘‘spliced’’ bundle with G×N(H)/H as the structure group has to be used. The unified space is then taken as the bundle space of its associated bundle.