A generalisation of g-rectifying and g-normal curves in Lorentzian n-space

In this paper, we introduce and analyze $g-$rectifying curves (spacelike and null curves) and $\ g-$normal curves in Lorentzian $n$-space, building upon the established notion of rectifying curves and normal curve, respectively. Our generalization extends this definition by considering an $$position vector field, $\xi _{g}(s)=\int g(s)d\xi$, where $g$ is an integrable function in the arc-length parameter $s$. An $g$-rectifying curves(or $g-$normal curves) are then defined as an arc-length parametrized curve $\xi$ in Lorentzian $n-$space such that its $g$-position vector consistently lies within its rectifying space(or normal space). The primary objective of this work is to provide a comprehensive characterization and classification of these $g$-rectifying curves and $g-$normal curves, thereby expanding the geometric understanding of curves in Lorentzian $n$-spaces.