## 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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