Abstract
Let x: M n−1 → Rn be an umbilical free hypersurface with non-zero principal curvatures. Two basic invariants of M under the Laguerre transformation group of Rn are Laguerre form C and Laguerre tensor L. In this paper, n > 3) complete hypersurface with vanishing Laguerre form and with constant Laguerre scalar curvature R in Rn, denote the trace-free Laguerre tensor by ˜\(\widetilde L = L - \frac{1}{{n - 1}}tr\left( L \right)\) · Id. If \(\widetilde L = L - \frac{1}{{n - 1}}tr\left( L \right)\), then M is Laguerre equivalent to a Laguerre isotropic hypersurface; and if \({\sup _M}\left\| {\widetilde L} \right\| = \frac{{\sqrt {\left( {n - 1} \right)\left( {n - 2} \right)} R}}{{\left( {n - 1} \right)\left( {n - 2} \right)\left( {n - 3} \right)}},\), M is Laguerre equivalent to the hypersurface ˜x: H 1 × S n−2 → Rn.
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The first author is supported by Scienticfic Research Fund of Yunnan Provincial Education Department (Grant No. 2014Y445) and Yunnan Applied Basic Research Young Project; the second author is supported by Scienticfic Research Fund of Yunnan Provincial Education Department (Grant No. 2015Y101)
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Fang, J., Li, F. Complete hypersurfaces with constant laguerre scalar curvature in ℝn . Acta. Math. Sin.-English Ser. 32, 715–724 (2016). https://doi.org/10.1007/s10114-016-4531-6
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DOI: https://doi.org/10.1007/s10114-016-4531-6
Keywords
- Laguerre isoparametric hypersurface
- Laguerre second fundamental form
- Laguerre metric
- Laguerre form
- parallel Laguerre form