On Level Hypersurfaces of the Vertical Lift of a Submersion

Year 2024, Volume: 16 Issue: 1, 272 - 284, 30.06.2024

Mehmet Yıldırım , Ayşenur Özkan

https://doi.org/10.47000/tjmcs.1397889

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.

Keywords

Level surfaces, tangent bundle, vertical lift.

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