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On Legendre curves in contact 3-manifolds. (English) Zbl 0799.53040

A \((2n+1)\)-dimensional manifold \(M\) is said to be a contact manifold if it carries a global 1-form \(\omega\) such that \(\omega\land(d\omega)^ n\neq 0\) everywhere. The \(2n\)-dimensional distribution \(D=\ker\omega\) is not integrable, the maximum dimension of an integral submanifold is \(n\). A 1-dimensional integral submanifold will be called Legendre curve, especially to avoid confusion with an integral curve of the characteristic vector field \(\xi\) of the contact structure \(\omega\). A contact manifold has an almost contact structure \((\phi,\xi,\omega)\) where \(\omega(\xi)=1\), \(\phi(\xi)=0\), \(\phi^ 2=-I_ d+\omega\otimes\xi\). Then an associated metric \(g\) (not unique) can be found such that \(\omega=g(\xi,\cdot)\) and \((d\omega)(X,Y)=g(X,\phi Y)\); \((M,\omega,g)\) is called contact metric manifold. If the natural almost complex structure induced on \(M\times{\mathbb{R}}\) is integrable, then the contact structure is called normal. A Sasakian manifold is a normal contact metric manifold. In this paper the authors study the Legendre curves, from the Riemannian point of view, in a contact metric 3- manifold. They obtain the following nice result: a contact metric 3- manifold is Sasakian if, and only if, every Legendre curve has torsion equal to 1. Moreover, the authors show that for the standard contact structure on \({\mathbb{R}}^ 3\) with its standard Sasakian metric the curvature of a Legendre curve is equal to twice the curvature of its projection to the \(xy\)-plane with respect to the Euclidean metric. Thus, they observe that this metric on \({\mathbb{R}}^ 3\) is more natural for the study of Legendre curves than the Euclidean metric. Finally Legendre curves of finite type are studied.
Reviewer: D.Perrone (Lecce)

MSC:

53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53C20 Global Riemannian geometry, including pinching
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References:

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