Extended cross product in a 3-dimensional almost contact metric manifold with applications to curve theory (Q2911330)

For an almost contact metric manifold \(M(\varphi,\xi,\eta,g)\), the author defines the vector cross product by NEWLINE\[NEWLINEX\wedge Y = \null-g(X,\varphi Y) -\eta(Y)\varphi X + \eta(X)\varphi Y,\quad X,Y\in\mathfrak X(M).NEWLINE\]NEWLINENEWLINENEWLINE(Reviewer's remark: This is the vector cross product arising from the standard volume element \((-3!\varPhi\wedge\eta)\) on such manifolds.)NEWLINENEWLINE The geometric properties of this cross product are derived. Next, this product is applied to the theory of curves in \(M(\varphi,\xi,\eta,g)\). Finally, the author recovers and extends some classical theorems about the curvature and the torsion of Legendre and slant curves in contact metric manifolds of dimension \(3\).