Showing papers by "David E. Blair published in 1994"

On Legendre curves in contact 3-manifolds

Abstract: It is first observed that on a 3-dimensional Sasakian manifold the torsion of a Legendre curve is identically equal to +1. It is then shown that, conversely, if a curve on a Sasakian 3-manifold has constant torsion +1 and satisfies the initial conditions at one point for a Legendre curve, it is a Legendre curve. Furthermore, among contact metric structures, this property is characteristic of Sasakian metrics. For the standard contact structure onR3 with its standard Sasakian metric the curvature of a Legendre curve is shown to be twice the curvature of its projection to thexy-plane with respect to the Euclidean metric. Thus this metric onR3 is more natural for the study of Legendre curves than the Euclidean metric.

When is the tangent sphere bundle

Abstract: In [2] one of the authors showed that the s tandard contact metric structure on the tangent sphere bundle is locally symmetric if and only if the base manifold is fiat or of dimension 2 and of constant curvature +1. In this paper we show that this structure is conformally flat if and only if the base manifold is a surface of constant Gaussian curvature 0 or -t--1. In the final section of the paper we give an additional result on conformally flat contact metric manifolds.