Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

Article

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Bull. Korean Math. Soc. 2019; 56(1): 229-243

Online first article September 6, 2018      Printed January 31, 2019

https://doi.org/10.4134/BKMS.b180188

Copyright © The Korean Mathematical Society.

Some integrations on null hypersurfaces in Lorentzian manifolds

Fortun\'{e} Massamba, Samuel Ssekajja

Private Bag X01, Scottsville 3209; Private Bag X01, Scottsville 3209

Abstract

We use the so-called pseudoinversion of degenerate metrics technique on foliated compact null hypersurface, $M^{n+1}$, in Lorentzian manifold $\overline{M}^{n+2}$, to derive an integral formula involving the $r$-th order mean curvatures of its foliations, $\mathcal{F}^{n}$. We apply our formula to minimal foliations, showing that, under certain geometric conditions, they are isomorphic to $n$-dimensional spheres. We also use the formula to deduce expressions for total mean curvatures of such foliations.

Keywords: null hypersurface, Newton transformation, foliation

MSC numbers: Primary 53C50; Secondary 53C12, 53C40

Supported by: This work is based on the research supported wholly/in part by the National Research Foundation of South Africa (Grant Numbers: 95931 and 106072).