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Slant curves in 3-dimensional normal almost contact geometry. (English) Zbl 1269.53020

The purpose of the paper is to begin the study of slant curves in normal almost contact metric manifolds. An almost contact structure on a manifold \(M\) consists of a 1-form \(\eta\), a vector field \(\xi\) and a field of endomorphisms \(\varphi\) such that \(\varphi^2=-I+\eta\otimes\xi\) and \(\eta(\xi)=1\). The product of \(M\) with the real line admits a natural almost complex structure; if this structure is integrable we say that the almost contact structure is normal. Given an almost contact structure there exists a Riemannian metric \(g\) such that \(g(\varphi X,\varphi Y)=g(X,Y)-\eta(X)\eta(Y)\) and we refer to an almost contact metric structure.
The paper studies curves in normal almost contact metric manifolds including the analogue of the Lancret invariant (the ratio of torsion to curvature for a curve in Euclidean 3-space). Define the structural angle \(\theta\) of a curve \(\gamma\) by \(\cos\theta(s)=g(\gamma',\xi)\). The curve is called a \(\theta\)-slant curve if \(\theta\) is a constant. If \(\theta={\pi\over 2}\) or \(\theta={3\pi\over 2}\), \(\gamma\) is a Legendre curve. The Lancret invariant of \(\gamma\) is defined as \({\cos\theta\over |\sin\theta|}\).
The authors first compute the curvature, torsion and the Lancret invariant of a \(\theta\)-slant curve in a 3-dimensional normal almost contact metric manifold. Then they introduce a second fundamental form and mean curvature vector field \(H\) for a curve. The mean curvature vector field is said to be proper if there exists a function \(\lambda\) along the curve such that \(\Delta H=\lambda H\). It is shown that a non-geodesic \(\theta\)-slant curve has a proper mean curvature vector field if and only if it is a helix. For examples the authors construct a class of normal almost contact metric manifolds and give examples of a non-geodesic \(\theta\)-slant curve which is a generalized helix and of a helix with proper mean curvature vector field.

MSC:

53B25 Local submanifolds
53D15 Almost contact and almost symplectic manifolds
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
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