Showing papers on "Frame bundle published in 1994"
161 citations
TL;DR: In this article, the Yang-Mills heat flow in a vector bundle over a compact Riemannian four-manifold for given initial connection of finite energy is established.
Abstract: Global existence and uniqueness is established for the Yang-Mills heat flow in a vector bundle over a compact Riemannian four-manifold for given initial connection of finite energy. Our results are analogous to those valid for the evolution of harmonic maps of Riemannian surfaces.
78 citations
TL;DR: In this paper, the authors describe the standard Liouville form θ and the symplectic form dθ in terms of the right Maurer Cartan form and the left moment mapping (of the right action of G on itself).
Abstract: On a cotangent bundle T*G of a Lie group G one can describe the standard Liouville form θ and the symplectic form dθ in terms of the right Maurer Cartan form and the left moment mapping (of the right action of G on itself), and also in terms of the left Maurer–Cartan form and the right moment mapping, and also the Poisson structure can be written in related quantities. This leads to a wide class of exact symplectic structures on T*G and to Poisson structures by replacing the canonical momenta of the right or left actions of G on itself by arbitrary ones, followed by reduction (to G cross a Weyl‐chamber, e.g.). This method also works on principal bundles.
40 citations
TL;DR: In this article, the authors established the relationship between n-symplectic geometry on the bundle of linear frames LM of an n-dimensional manifold M and canonical symplectic geometry of polynomial observables on the cotangent bundle T ∗M.
36 citations
TL;DR: In this paper, a transitive action of a symmetry group G on the base space B of a non-Abelian principal bundle P ω with a connection ω was studied.
15 citations
11 citations
01 Jan 1994
TL;DR: In this paper, Ghione et al. investigated the restriction of the tangent bundle of IP to a curve X ⊂ IP and showed that the restricted bundle is semistable and moreover simple if g ≥ 2 if the degree is high with respect to the genus.
Abstract: The purpose of this paper is to investigate the restriction of the tangent bundle of IP to a curve X ⊂ IP. The corresponding question for rational curves was investigated by L. Ramella [7] and F. Ghione, A. Iarrobino and G. Sacchiero [2] in the case of rational curves. Let us also mention that D. Laksov [6] proved that the restricted tangent bundle of a projectivly normal curve does not split unless the curve is rational. We will show the following theorem (See 3.1): Theorem In the variety of smooth connected space curves of genus g ≥ 1 and degree d ≥ g+ 3 there exists a nonempty dense open subset where the restricted tangent bundle is semistable and moreover simple if g ≥ 2 If the degree is high with respect to the genus (d > 3g), we get a postulation formula for the strata with a given Harder-Narasimhan polygon, following results of R. Hernandez [5]. In case of plane curves the situation is simpler due to Theorem If X is a smooth plane curve of degree d, the restricted tangent bundle is stable for d ≥ 3, of splitting type (3, 3) for a conic, and of splitting type (2, 1) for a line. Proof: (following D. Huybrechts) We denote by E the tangent bundle of IP 2 twisted by OIP 2(−1). We first suppose that d > 2. We use the facts: 1. E is stable, c1(E) = 1 and c2(E) = 1.
10 citations
TL;DR: In this article, the determinant line bundle over moduli space of stable bundles on abelian surfaces is studied and the authors evaluate their analytic torsions and prove that the Mukai transform preserves the line bundle as a hermitian line bundle.
Abstract: We study the determinant line bundle over moduli space of stable bundles on abelian surfaces. We evaluate their analytic torsions. We extend Mukai's version of the Parseval Theorem to L 2 metrics on cohomology groups. We prove that the Mukai transform preserves the determinant line bundle as a hermitian line bundle. This is done by induction via the natural boundary of the moduli spaces.
7 citations
Abstract: In the Finsler-spacetime tangent bundle, a simple solution is determined to the torsion relations that were obtained previously to maintain (1) compatibility with Cartan's theory of Finsler space, (2) the almost complex structure, and (3) the vanishing of the covariant derivative of the almost complex structure.
7 citations
TL;DR: In this article, a complete set of invariants for non-degenerate oriented surfaces M 2 in ℝ4 relative to the action of the general affine group on M 2 were derived.
Abstract: Employing the method of moving frames, i.e. Cartan's algorithm, we find a complete set of invariants for nondegenerate oriented surfacesM2 in ℝ4 relative to the action of the general affine group on ℝ4. The invariants found include a normal bundle, a quadratic form onM2 with values in the normal bundle, a symmetric connection onM2 and a connection on the normal bundle. Integrability conditions for these invariants are also determined. Geometric interpretations are given for the successive reductions to the bundle of affine frames overM2, obtained by using the method of moving frames, that lead to the aforementioned invariants. As applications of these results we study a class of surfaces known as harmonic surfaces, finding for them a complete set of invariants and their integrability conditions. Further applications involve the study of homogeneous surfaces; these are surfaces which are fixed by a group of affine transformations that act transitively on the surface. All homogeneous harmonic surfaces are determined.
7 citations
TL;DR: In this article, it was proved that a differential form on T*(M) is invariant under this representation (resp. gauge invariant) if and only if it belongs to the R-algebra generated by the canonical symplectic form of the cotangent bundle.
Abstract: The cotangent bundle of a manifold M can be identified with the bundle of connections of the trivial bundle M×U(1). Hence, there exists a natural representation of the U(1)‐invariant vector fields on M×U(1) extending the gauge representation into the Lie algebra of vector fields on T*(M). It is proved herein that a differential form on T*(M) is invariant under this representation (resp. gauge invariant) if and only if it belongs to the R‐algebra [resp. to the Ω (M)‐algebra] generated by the canonical symplectic form of the cotangent bundle.
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TL;DR: In this paper, the authors introduced the notion of continuous twisted partial actions of a locally compact group on a C*-algebra and constructed an associated C*algebraic bundle called the semidirect product bundle.
Abstract: We introduce the notion of continuous twisted partial actions of a locally compact group on a C*-algebra. With such, we construct an associated C*-algebraic bundle called the semidirect product bundle. Our main theorem shows that, given any C*-algebraic bundle which is second countable and whose unit fiber algebra is stable, there is a continuous twisted partial action of the base group on the unit fiber algebra, whose associated semidirect product bundle is isomorphic to the given one.
01 Jan 1994
TL;DR: In this article, the degree of the top Segre class of a standard rank-4 vector bundle e D 4 on the Hilbert scheme H 4 = Hilb 4 S of 0-dimensional subschemes of length 4 on a smooth projective surface S.
Abstract: This work is a continuation of the paper [4] of the present collection. Using the results of [4] (and keeping the notations introduced there), we compute here the degree \({\delta _4} = \int\limits_{{H_4}} {{s_8}(\varepsilon _D^4)} \) of the top Segre class of a standard rank-4 vector bundle e D 4 on the Hilbert scheme H 4 = Hilb 4 S of 0-dimensional subschemes of length 4 on a smooth projective surface S.
TL;DR: In this article, the authors give a characterization of ampleness of the cotangent bundle of a branched covering of compact complex surfaces, where the ramification set in X consists of smooth curves meeting with at most normal crossings.
Abstract: Let /: X —> Y be a branched covering of compact complex surfaces, where the ramification set in X consists of smooth curves meeting with at most normal crossings and Y has ample cotangent bundle. We further assume that / is locally of form (u,υ) —> (un ,vm). We characterize ampleness of T*X . A class of examples of such X, which are branched covers of degree two, is provided. 1. Introduction. An interesting problem in surface theory is the construction and characterization of surfaces with ample cotangent bundle. They are necessarily algebraic surfaces of general type. Natural examples occur among the complete intersection surfaces of abelian varieties. More subtle examples are those constructed by Hirzebruch [6] using line-arrangements in the plane. The characterization of those of Hirzebruch's line-arrangement surfaces with ample cotangent bundle is due to Sommese [8]. In this article, we will give a characterization of ampleness of the cotangent bundle of a class of surfaces which branch cover another surface with ample cotangent bundle. We will also construct certain branched coverings of explicit line-arrangement surfaces; these constructions will again have ample cotangent bundle. For any vector bundle E over a base manifold M, the projectivization P(E) is a fiber bundle over M, with fiber Pq(E) over q e M given by T?g(E) « (2?*\0)/C*. There is a tautological linebundle ξβ over P(E) satisfying (i) ζE\F ^ « 0(1)P ^ V
01 Jan 1994
TL;DR: The geometry of the total space of the tangent bundle to a smooth manifold is very rich and contains a lot of geometrical objects of theoretical interest and of a great importance in constructing of various geometry models useful in Physics.
Abstract: The geometry of the total space of the tangent bundle to a smooth manifold is very rich. It contains a lot of geometrical objects of theoretical interest and of a great importance in constructing of various geometrical models useful in Physics. For the similar reasons the geometry of the total space of a vector bundle was intensively studied in the last fifteen years.
TL;DR: In this article, the authors show that the Segre polynomial determines the minimal number of sections spanning a vector bundle spanned by global sections, i.e., the minimum number of segments spanning the vector.
Abstract: We show that the Segre polynomial determines the minimal number of sections spanning a vector bundle spanned by global sections.
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TL;DR: In this article, the structure group of the principal linear frame bundle of the fermion-gravitation complex is reduced to the Lorentz group, and the corresponding momenta correspond to derivatives of fields with respect to all world coordinates, not only time.
Abstract: Gravitation theory meets spontaneous symmetry breaking when the structure group of the principal linear frame bundle $LX$ over a world manifold $X^4$ is reducible to the Lorentz group $SO(3,1)$. The physical underlying reason of this reduction is Dirac fermion matter possessing only exact Lorentz symmetries. The associated Higgs field is a tetrad gravitational field $h$ represented by a section of the quotient $\Si$ of $LX$ by $SO(3,1)$. The feature of gravity as a Higgs field issues from the fact that, in the presence of different tetrad fields, there are nonequivalent representations of cotangent vectors to $X^4$ by Dirac's matrices. It follows that fermion fields must be regarded only in a pair with a certain tetrad field. These pairs constitute the so-called fermion-gravitation complex and are represented by sections of the composite spinor bundle $S\to\Si\to X^4$ where values of tetrad gravitational fields play the role of coordinate parameters, besides familiar world coordinates. In Part I of the work [gr-qc:9405013], geometry of this composite spinor bundle has been investigated. This Part is devoted to dynamics of the fermion-gravitation complex. It is a constraint system to describe which we use the covariant multimomentum Hamiltonian formalism when canonical momenta correspond to derivatives of fields with respect to all world coordinates, not only time. On the constraint space, the canonical momenta of tetrad gravitational fields are equal to zero, otherwise in the presence of fermion fields.
TL;DR: Alamo and Gbmez as discussed by the authors proved that a G-invariant neighbourhood of the singular set C in a manifold M is completely determined by the G-vector bundle restriction of the tangent bundle of M to Z.