Showing papers on "Unit tangent bundle published in 1986"
01 Jan 1986
19 citations
01 Apr 1986
TL;DR: In this paper, Eisenbud et al. studied the relationship between the degrees of the line bundles of the splitting of O*Tp3 and the geometry of the curve C.
Abstract: Let O*Tp3 be the pull-back of the tangent bundle to P3 via a parametrization ik of a rational, reduced, irreducible curve C in P3 contained in an irreducible quadric surface. Since C is rational, the bundle i*Tp3 splits into the direct sum of three line bundles. In this paper we study the relationship between the degrees of the line bundles of the splitting of O*Tp3 and the geometry of the curve C. 0. Introduction. Let Tpr be the tangent bundle to the r-dimensional projective space pr over an algebraically closed field. Throughout this paper C will denote a rational, reduced, irreducible curve in pr of degree dc not contained in any hyperplane and 4: p' -l pr will be a parametrization of C. We will drop the subscript C anytime this will not lead to confusion. We consider the vector bundle 4'*Tpr on P' and, by abuse of language, we will refer to it as the tangent bundle restricted to the curve C. By a well-known theorem of Grothendieck V)*Tpr splits into the direct sum of r line bundles. The aim of this paper is to determine the decomposition of the tangent bundle restricted to a singular curve lying on a quadric surface in P3. The motivation for studying V)*Tpr comes from the relationship (suggested in [EV]) between it and the geometry of the embedded curve, and because of the information about the normal bundle, the Hilbert function, and the regularity of the curve one can derive from the splitting of 4'*Tpr. We distinguish between the cases where the quadric surface Q is singular from the ones where it is not. If C lies on a quadric cone then we will denote by a the intersection number of C with the lines of the ruling on Q outside the vertex; if C lies on a smooth quadric then (a, d a) will be its divisor class. In both cases we can assume that a < [d/2J. Our main results are the following THEOREM (0. 1). If C lies on a quadric cone, it passes through its vertex and one of the following holds: (i) a < Ld/3], (ii) Ld/3j < a and there exist two points on C of multiplicity a, (iii) Ld/3J < a and there exists a point on C of multiplicity a such that the total number of inflections at the point is a. Then O*Tpe v Opi (d + a) ED Opi (d + a) (D Op (2d 2a). Received by the editors November 4, 1985. 1980 Mathemotics Sukiect Cla&sifintion (1985 Revision). Primary 14H45. This paper is part of the author's Ph.D. thesis prepared under the supervision of Dr. D. Eisenbud (Brandeis University, May 1985). (g)1986 American Mathematical Society 0002-9939/86 $1.00 + $.25 per page
14 citations
TL;DR: In this article, a notion of equivariant index is introduced to describe the behavior of tangent G-vector fields on smooth G-manifolds near isolated zeros, and the monoid of G-homotopy classes of self-maps of the unit sphere S (V ) in a real orthogonal (finite dimensional) G-module V, which is the unstable analogue of a classical result of Segal.
9 citations
7 citations
TL;DR: In this article, Lagrangian dynamical systems on tangent bundles of differentiable manifolds are considered, and necessary and sufficient conditions for finding a function on the base manifold whose time derivative, if added to the gauge invariant Lagrangians, yields a strictly invariant one.
Abstract: We consider Lagrangian dynamical systems on tangent bundles of differentiable manifolds whose Lagrangian functions are gauge invariant under the action of a Lie group on the base manifold. We then obtain necessary and sufficient conditions for finding a function on the base manifold whose time derivative, if added to the gauge-invariant Lagrangian, yields a strictly invariant one. The problem is transported from the tangent bundle also to the cotangent bundle.
6 citations
01 Jan 1986
3 citations
TL;DR: In this article, Pseudo-Sasakian manifolds M(U,E,,g) endowed with a contact conformal connection are defined, and it is proved that these manifolds are space forms M(K),K < O, and somo remarkable properttos of the 1,ie algebra of infinitesimal transformatton.
Abstract: ABS’I’RACI’. Pseudo-Sasakian manifolds M(U,E,,,g) endowed wlth a contact conformal connection are defined. It is proved tlat sucl manifolds are space forms M(K),K < O, and somo remarkable properttos of the 1,ie algebra of infinitesimal transformatton. of the principal vector feld U on M are discussed. Properties of tle leaves of a co-tsotroptc foliation on I’! and properties of the tangent bundle manifold TM having lq as a basis nro studied.
1 citations
TL;DR: In this article, a characterization of 2-dimensional Riemannian manifolds (M, g) with constant Gaussian curvature is presented, and it is shown that these manifolds admit an almost-regular vector field belonging to an eigenspace of the Ricci operator.
Abstract: Characterisation of 2-dimensional Riemannian manifolds (M, g) (in particular, of surfaces with constant gaussian curvatureK=1/c2, o,−1/c2, respectively) whose tangent circle bundle (TcM, gs) (gs=Sasaki metric) admit an «almost-regular» vector field belonging to an eigenspace of the Ricci operator.
1 citations
TL;DR: In this article, it was shown that the third Chern-number of a semistable Rk-3 vector bundle on a smooth hypersurface of degree 3 in ℙ4 can be bounded by the first and second Chern numbers of the bundle.
Abstract: It is shown that the third Chern-number of a semistable Rk-3-vector bundle on a smooth hypersurface of degree 3 in ℙ4 can be bounded by the first and the second Chern-number of the bundle.
TL;DR: In this article, the fiber vierbein (frame) is considered as dependent indirectly on an element / of the fiber group, which we choose as a harmonic mapping from the basis manifold.
Abstract: Fibre bundle formulations of field theory are considered approximately by the appropriate vector bundle. The fibre vierbein (frame) is considered as dependent indirectly on an element / of the fibre group, which we choose as a harmonic mapping from the basis manifold. We do the same for the other fundamental geometrical quantity called connection, but choose another independent harmonic mappingh instead of /. Additionally, we generalize the torsion-free condition to this more general vector bundle other than tangent bundle, by using the pullback mapping. A new gauge-invariant Lagrangian is proposed.