Minimal immersions of Riemannian manifolds in products of space forms. (English) Zbl 1305.53064
Summary: In this paper, we give natural extensions to cylinders and tori of a classical result due to T. Takahashi [J. Math. Soc. Japan 18, 380–385 (1966; Zbl 0145.18601)] about minimal immersions into spheres. More precisely, we deal with Euclidean isometric immersions whose projections in \(\mathbb R^N\) satisfy a spectral condition of their Laplacian.
MSC:
| 53C42 | Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) |
Citations:
Zbl 0145.18601References:
| [1] | Alías, L. J.; Ferrández, A.; Lucas, P., Hypersurfaces in space forms satisfying the condition \(\Delta x = A x + B\), Trans. Amer. Math. Soc., 347, 1793-1801 (1995) · Zbl 0829.53050 |
| [2] | Chen, B. Y., Total Mean Curvature and Submanifolds of Finite Type (1984), World Scientific Publ.: World Scientific Publ. New Jersey · Zbl 0537.53049 |
| [3] | Eells, J.; Sampson, J. H., Harmonic mapping of Riemannian manifold, Amer. J. Math., 186, 109-160 (1964) · Zbl 0122.40102 |
| [4] | Garay, O. J., An extension of Takahashi’s theorem, Geom. Dedicata, 34, 105-112 (1990) · Zbl 0704.53044 |
| [5] | Lira, J. H.; Tojeiro, R.; Vitório, F., A Bonnet theorem for isometric immersions into products of space forms, Arch. Math., 95, 469-479 (2010) · Zbl 1208.53061 |
| [6] | Park, J., Hypersurfaces satisfying the equation \(\Delta x = R x + b\), Proc. Amer. Math. Soc., 120, 317-328 (1994) · Zbl 0819.57005 |
| [7] | Simons, J., Minimal varieties in Riemannian manifolds, Ann. of Math., 88, 62-105 (1968) · Zbl 0181.49702 |
| [8] | Takahashi, T., Minimal immersions of Riemannian manifolds, J. Math. Soc. Japan, 18, 380-385 (1966) · Zbl 0145.18601 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
