The present paper is devoted to study the curvature and torsion of Frenet Legendre curves in 3-dimensional normal almost paracontact metric manifolds. Moreover, in this class of manifolds, properties of non-Frenet Legendre curves (with null tangents or null normals or null binormals) are obtained. Many examples of Legendre curves are constructed.Some of the present results are analogous to those obtained by the author in  for Frenet Legendre curves in 3-dimensional normal almost contact manifolds.
Mathematics Subject Classification (2000). 53D15, 53C25.
The present paper is devoted to study the curvature and torsion of slant Frenet curves in 3-dimensional normal almost paracontact metric manifolds. Moreover, in this class of manifolds, properties of non-Frenet slant curves (with null tangents or null normals) are studied. We illustrate such results by some examples.
Almost paracontact metric manifolds are the famous examples of almost para-CR manifolds. We find necessary and sufficient conditions for such manifolds for be para-CR. Next we examine these conditions in certain subclasses of almost paracontact metric manifolds. Especially, it is shown that the normal almost paracontact metric manifolds are para-CR. We establish necessary and sufficient conditions for paracontact metric manifolds as well as for almost paracosymplectic manifolds to be para-CR. We find also basic curvature identities for para-CR paracontact metric manifolds and study their consequences. Among others, we prove that any para-CR paracontact metric manifold of constant sectional curvature and of dimension greather than tree must be para-Sasakian and its curvature equal to minus one. The last assertion do not hold in dimension tree. Moreover, we show that a conformally flat para-Sasakian manifold is of constant sectional curvature equal to minus one. New classes of examples of para-CR manifolds are established.Keywords: Para-CR manifolds; para-Sasakian manifolds; almost paracontact metric manifolds; paracontact metric manifolds, almost para-cosymplectic manifolds. AMS Subject Classification: 53C15, 53C50.
PreliminariesLet M be an almost paracontact manifold and (ϕ, ξ, η) its almost paracontact structure (e.g. [9,17]). This means that M is an (2n + 1)-dimensional differentiable manifold and ϕ, ξ, η are tensor fields on M of type (1, 1), (1, 0), (0, 1), respectively, such thatMoreover the tensor field ϕ induces an almost paracomplex structure on the paracontact distribution D = Ker η, i.e. the eigendistributions D ± corresponding to the eigenvalues ±1 of ϕ are both n-dimensional. A pseudo-Riemannian metric g on M satisfying the conditionis said to be compatible with the structure (ϕ, ξ, η). If g is such a metric, then the quadruplet (ϕ, ξ, η, g) is called an almost paracontact metric structure and M an almost paracontact metric manifold. For such a manifold, we additionally have η(X) = g(X, ξ), and we define the (skew-symmetric) fundamental 2-form Φ by Φ(X, Y ) = g(X, ϕY ).
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