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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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