Hypersurfaces in space forms satisfying the condition Δ𝑥=𝐴𝑥+𝐵

Alías, Luis J.; Ferrández, Angel; Lucas, Pascual

doi:10.1090/S0002-9947-1995-1257095-8

Hypersurfaces in space forms satisfying the condition $\Delta x=Ax+B$
HTML articles powered by AMS MathViewer

by Luis J. Alías, Angel Ferrández and Pascual Lucas PDF

Trans. Amer. Math. Soc. 347 (1995), 1793-1801 Request permission

Abstract:

In this work we study and classify pseudo-Riemannian hypersurfaces in pseudo-Riemannian space forms which satisfy the condition $\Delta x = Ax + B$, where $A$ is an endomorphism, $B$ is a constant vector, and $x$ stands for the isometric immersion. We prove that the family of such hypersurfaces consists of open pieces of minimal hypersurfaces, totally umbilical hypersurfaces, products of two nonflat totally umbilical submanifolds, and a special class of quadratic hypersurfaces.

References

Luis J. Alías, Angel Ferrández, and Pascual Lucas, Surfaces in the $3$-dimensional Lorentz-Minkowski space satisfying $\Delta x=Ax+B$, Pacific J. Math. 156 (1992), no. 2, 201–208. MR 1186802, DOI 10.2140/pjm.1992.156.201
Luis J. Alías, Angel Ferrández, and Pascual Lucas, Submanifolds in pseudo-Euclidean spaces satisfying the condition $\Delta x=Ax+B$, Geom. Dedicata 42 (1992), no. 3, 345–354. MR 1164541, DOI 10.1007/BF02414072
Bang-Yen Chen, Finite-type pseudo-Riemannian submanifolds, Tamkang J. Math. 17 (1986), no. 2, 137–151. MR 872686
Bang-Yen Chen and Mira Petrovic, On spectral decomposition of immersions of finite type, Bull. Austral. Math. Soc. 44 (1991), no. 1, 117–129. MR 1120399, DOI 10.1017/S0004972700029518
Franki Dillen, Johan Pas, and Leopold Verstraelen, On surfaces of finite type in Euclidean $3$-space, Kodai Math. J. 13 (1990), no. 1, 10–21. MR 1047590, DOI 10.2996/kmj/1138039155
Jörg Hahn, Isoparametric hypersurfaces in the pseudo-Riemannian space forms, Math. Z. 187 (1984), no. 2, 195–208. MR 753432, DOI 10.1007/BF01161704
Th. Hasanis and Th. Vlachos, Hypersurfaces of $E^{n+1}$ satisfying $\Delta x=Ax+B$, J. Austral. Math. Soc. Ser. A 53 (1992), no. 3, 377–384. MR 1187856
H. Blaine Lawson Jr., Local rigidity theorems for minimal hypersurfaces, Ann. of Math. (2) 89 (1969), 187–197. MR 238229, DOI 10.2307/1970816
Martin A. Magid, Isometric immersions of Lorentz space with parallel second fundamental forms, Tsukuba J. Math. 8 (1984), no. 1, 31–54. MR 747443, DOI 10.21099/tkbjm/1496159942
Joonsang Park, Hypersurfaces satisfying the equation $\Delta x=Rx+b$, Proc. Amer. Math. Soc. 120 (1994), no. 1, 317–328. MR 1189750, DOI 10.1090/S0002-9939-1994-1189750-7
Patrick J. Ryan, Homogeneity and some curvature conditions for hypersurfaces, Tohoku Math. J. (2) 21 (1969), 363–388. MR 253243, DOI 10.2748/tmj/1178242949