Hypersurfaces satisfying the equation Δ𝑥=𝑅𝑥+𝑏

Park, Joonsang

doi:10.1090/S0002-9939-1994-1189750-7

Hypersurfaces satisfying the equation $\Delta x=Rx+b$
HTML articles powered by AMS MathViewer

by Joonsang Park PDF

Proc. Amer. Math. Soc. 120 (1994), 317-328 Request permission

Abstract:

We prove that a hypersurface in a space form or in Lorentzian space whose immersion $x$ satisfies $\Delta x = Rx + b$ is minimal or isoparametric. In particular, we locally classify such hypersurfaces which are not minimal.

References

Surfaces in the

dimensional space satisfying

E. Cartan, Sur les variétés de courbure constante d’un espace euclidien ou non-euclidien, Bull. Soc. Math. France 47 (1919), 125–160 (French). MR 1504786, DOI 10.24033/bsmf.997
E. Cartan, Sur les variétés de courbure constante d’un espace euclidien ou non-euclidien, Bull. Soc. Math. France 48 (1920), 132–208 (French). MR 1504796, DOI 10.24033/bsmf.1007
Oscar J. Garay, An extension of Takahashi’s theorem, Geom. Dedicata 34 (1990), no. 2, 105–112. MR 1061282, DOI 10.1007/BF00147319
Martin A. Magid, Lorentzian isoparametric hypersurfaces, Pacific J. Math. 118 (1985), no. 1, 165–197. MR 783023, DOI 10.2140/pjm.1985.118.165
Hans Friedrich Münzner, Isoparametrische Hyperflächen in Sphären, Math. Ann. 251 (1980), no. 1, 57–71 (German). MR 583825, DOI 10.1007/BF01420281
Barrett O’Neill, Semi-Riemannian geometry, Pure and Applied Mathematics, vol. 103, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983. With applications to relativity. MR 719023
Richard S. Palais and Chuu-Lian Terng, Critical point theory and submanifold geometry, Lecture Notes in Mathematics, vol. 1353, Springer-Verlag, Berlin, 1988. MR 972503, DOI 10.1007/BFb0087442

Famiglie di ipersuperifici isoparametriche neglie spazi euclidei ad un qualunque numero di dimension

27

Tsunero Takahashi, Minimal immersions of Riemannian manifolds, J. Math. Soc. Japan 18 (1966), 380–385. MR 198393, DOI 10.2969/jmsj/01840380

Lorentzian isoparametric submanifolds