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.

Keywords

• Level surfaces
• tangent bundle
• vertical lift.

References

1. Abraham, R., Marsden, J.E., Ratiu, T., Manifolds, Tensor Analysis and Applications, Springer Verlag, New York Inc., 1998.
2. Barletta, E., Dragomir, S., Duggal, K. L., Lightlike Foliations of Semi-Riemannian Manifolds, American Mathematical Society, Providence, RI, 2007.
3. Bejancu, A., Duggal, K.L., Lightlike submanifolds of Semi- Riemannian manifolds, Acta Appl. Math., 38 (1995), 197–215.
4. Brickell, F., Clark, R.S., Differentiable Manifolds, Van Nostrand Reinhold Company London, 1970.
5. Duggal, K.L., Bejancu, A., Lightlike Submanifolds of Semi-Riemannian Manifolds and Applications, Kluwer Academic Publishers, Dordrecht, 1996.
6. Güneş, R., Şahin, B., Kılıç, E., On Lightlike hypersurfaces of a semi-Riemannian manifold, Turk J Math., 27(2003), 283–297.
7. Massamba, F., Lightlike hypersurfaces of indefinite Sasakian manifolds with parallel symmetric bilinear forms, Differential Geometry - Dynamical Systems, 10(2008), 226–234.
8. Sahin, B., Gunes, R., Lightlike real hypersurfaces of indefinet quaternion Kaehler manifolds, J. Geometry, 75(2002), 151–165.

9. Sakaki, M., On the definition of minimal lightlike submanifolds, International Electronic Journal of Geometry, 3(1)(2010), 16–23.
10. Tani, M., Prolongations of hypersurfaces to tangent bundles, Kodai Math. Sem. Rep., 21(1969), 85–96.
11. Yano, K., Ishihara S., Tangent and Cotangent Bundles, Marcel Dekker Inc., New York 1973.
12. Yano, K., Kobayashi, S., Prolongations of tensor fields and connections to tangent bundles I, General Theory, Jour. Math. Soc. Japan, 18(1966), 194–210.
13. Yıldırım, M., On level hypersurfaces of the complete lift of a submersion, An. Ş t. Univ. Ovidus Constanta, 17(2)(2009), 231–252.