On Level Hypersurfaces of the Vertical Lift of a Submersion

Abstract

Suppose that $(M,G)$ be a Riemannian manifold and $f:M\rightarrow \mathbb{R}$ be a submersion. Then, the vertical lift of $f,$ $f^{v}:TM\rightarrow \mathbb{R}$ defined by $f^{v}=f\circ \pi $ is also a submersion. This interesting case, differently from [10], leads us to investigation of the level hypersurfaces of $f^{v}$ in tangent bundle $TM$. In this paper we obtained some differential geometric relations between level hypersurfaces of $f$ and $f^{v}.$ In addition, we noticed that, unlike [13], a level hypersurface of $f^{v}$ is always lightlike, i.e., it doesn't depend on any additional condition.