Showing papers on "Frame bundle published in 1975"

Stable principal bundles on a compact Riemann surface

Abstract: A radiator device adapted for attachment to a radiator pipe, comprising a mounting base wrapped about the radiator pipe and secured thereto by wire fastenings; a plurality of outwardly disposed radiating fins circularly positioned about said mounting base and extending outwardly therefrom. This radiating device is adapted to be installed upon any size flue pipe extending from a furnace to a chimney to transmit flue pipe heat to the surrounding atmosphere.

Negative vector bundles and complex Finsler structures

Abstract: A complex Finsler structure F on a complex manifold M is a function on the tangent bundle T(M) with the following properties. (We denote a point of T(M) symbolically by (z, ζ), where z represents the base coordinate and ζ the fibre coordinate.)

Isolated invariant sets for flows on vector bundles

Abstract: This paper studies isolated invariant sets for linear flows on the projective bundle associated to a vector bundle, e. g., the projective tangent flow to a smooth flow on a manifold. It is shown that such invariant sets meet each fiber, roughly in a disjoint union of linear subspaces. Isolated invariant sets which are intersections of attractors and repellers (Morse sets) are discussed. We show that, over a connected chain recurrent set in the base space, a Morse filtration gives a splitting of the projective bundle into a direct sum of invariant subbundles. To each factor in this splitting there corresponds an inferval of real numbers (disjoint from those for other factors) which measures the exponential rate of growth of the orbits in that factor. We use these results to see that, over a connected chain recurrent set, the zero section of the vector bundle is isolated if and only if the flow is hyperbolic. From this, it follows that if no equation in the hull of a linear, almost periodic differential equation has a nontrivial bounded solution then the solution space of each equation has a hyperbolic splitting.

The topology of group extensions of C systems

Abstract: The paper is concerned with the topological and metric properties of group extensions of C systems. The basic theorem describes the topologically transitive component, the ergodic component, and the K component of a group extension of a C system. It is shown that each of these components is a group sub-bundle of a principal bundle in which the group extension acts. The frame flow on a manifold of negative curvature is seen to be a special case of a group of extension of a C system. It is shown that the space of frames on a compact three-dimensional manifold with negative curvature does not have any group sub-bundles, so that the frame flow on manifolds of this class is topologically transitive, ergodic, and a K system.

G‐Hilbert bundles

Abstract: A notion of Hilbert bundle is proposed which leads to the construction of a ’’big’’ Hilbert space H starting from a family of Hilbert spaces. For this, such a family is equipped with a suitable structure, called Borel field structure. A meaningful relationship is established between the Borel structures which can be defined on the union of the Hilbert spaces of the family and the Borel field structures with which the family can be equipped. For a topological group G, the structure of G‐Hilbert bundle is defined linking in a suitable way a Hilbert bundle with actions of G. In the framework of a G‐Hilbert bundle, a continuous unitary representation of G in H can be constructed. The transitive G‐Hilbert bundles which are often used in the theory of induced representations of groups are shown to be a subclass of the class of the G‐Hilbert bundles which are proposed in this paper.

The structural consideration of the tangent bundleT(M n ) withg M over a riemannian manifoldM n

Abstract: A refinement of the geometry of tangent bundle[9] is made by presenting the proposition((2.3)) on tensor fields of a tangent bundle and it is shown that the Riemann metric gM of a tangent bundle previously called the Sasaki lift is nothing but the direct sum of the vertical and horizontal lift of the Riemann metric defined on the base Riemann manifold. The geometric meaning of the unit tensor field and the almost complex structure is given on the basis of the proposition((2.3)). By means of B. O’Neill’s scheme and of Y. Muto’s notion the geometry of horizontal and vertical distribution is developed and it is shown that the fibre is totally geodesic while the horizontal distribution admits the second fundamental tensor field which is skew-symmetric. In Y. Muto’s sense the tangent bundle with gM is an isometric and parallel fibred space.

On an invariant of a bundle of quadrics

Abstract: It is shown how the Hesse curve and theta-characteristic of a bundle of quadrics can be used to reconstruct the initial bundle. The corresponding construction is used to prove the global Torelli theorem for the intersection of three quadrics.Bibliography: 2 items