## On Legendre curves in 3-dimensional normal almost paracontact metric manifolds.(English)Zbl 1180.53080

Let $$M$$ be an almost paracontact manifold with a structure $$\Sigma=(\varphi,\xi,\eta)$$, where a $$1$$-form $$\eta$$, a vector field $$\xi$$, and a field of endomorphisms $$\varphi$$ satisfy $$\varphi^2=Id-\eta\otimes\xi$$ and $$\eta(\xi)=1$$. If there is a pseudo-Riemannian metric $$g$$ satisfying $$g(\varphi X,\varphi Y)=-g(X,Y)+\eta(X)\eta(Y)$$, then $$(M,\Sigma)$$ is called an almost paracontact metric manifold. If $$N_\varphi(X,Y)-2d\eta(X,Y)\xi=0$$, where $$N_\varphi$$ is the Nijenhuis tensor of $$\varphi$$, then $$(M,\Sigma)$$ is a normal almost paracontact metric manifold. A normal almost paracontact metric manifold $$M$$ is a para-Sasakian manifold if $$\Phi=d\eta$$, where $$\Phi(X,Y)=g(X,\varphi Y)$$. Let $$(M,g)$$ be a $$3$$-dimensional pseudo-Riemannian manifold. A curve $$\gamma:I\to M$$ such that $$g(\dot\gamma,\dot\gamma)=\pm1$$ is a Frenet curve on $$M$$ if $$\gamma$$ is of osculating order $$1$$, $$2$$, or $$3$$. If $$M$$ is almost paracontact metric, then $$\gamma$$ is a Legendre curve if $$\eta(\dot\gamma)=0$$. This interesting paper is devoted to study the curvature and torsion of Frenet Legendre curves in 3-dimensional normal almost paracontact metric manifolds. Some examples illustrating the results are presented.

### MSC:

 53D15 Almost contact and almost symplectic manifolds 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
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