Geodesic
Minimal hypersurfaces in $\mathbb{S}^5$ with vanishing Gauss--Kronecker curvature
Izvestiya. Mathematics , Tome 82 (2018) no. 3, pp. 477-493.

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We give a local description of complete minimal hypersurfaces in $\mathbb{S}^5$ with zero Gauss–Kronecker curvature, zero $3$-mean curvature and nowhere-zero second fundamental form.
Keywords: Gauss–Kronecker curvature, complete hypersurfaces.
Mots-clés : minimal hypersurfaces 
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M. M. Diniz; J. A. Vilhena; J. F. Zapata. Minimal hypersurfaces in $\mathbb{S}^5$ with vanishing Gauss--Kronecker curvature. Izvestiya. Mathematics , Tome 82 (2018) no. 3, pp. 477-493. http://geodesic.mathdoc.fr/item/IM2_2018_82_3_a1/

[1] S. S. Chern, M. do Carmo, S. Kobayashi, “Minimal submanifolds of a sphere with second fundamental form of constant length”, Functional analysis and related fields (Proc. Conf. for M. Stone, Univ. Chicago, Chicago, Ill., 1968), Springer, New York, 1970, 59–75 | DOI | MR | Zbl

[2] S. C. de Almeida, F. G. B. Brito, “Closed $3$-dimensional hypersurfaces with constant mean curvature and constant scalar curvature”, Duke Math. J., 61:1 (1990), 195–206 | DOI | MR | Zbl

[3] E. Cartan, “Sur des familles remarquables d'hypersurfaces isoparamétriques dans les espaces sphériques”, Math. Z., 45 (1939), 335–367 | DOI | MR | Zbl

[4] S. C. de Almeida, F. G. B. Brito, L. A. M. de Sousa Jr., “Closed hypersurfaces of $\mathbb S^4$ with two constant curvature functions”, Results Math., 50:1-2 (2007), 17–26 | DOI | MR | Zbl

[5] S. C. de Almeida, F. G. B. Brito, “Minimal hypersurfaces of $S^4$ with constant Gauss–Kronecker curvature”, Math. Z., 195:1 (1987), 99–107 | DOI | MR | Zbl

[6] J. Ramanathan, “Minimal hypersurfaces in $S^4$ with vanishing Gauss–Kronecker curvature”, Math. Z., 205:4 (1990), 645–658 | DOI | MR | Zbl

[7] Th. Hasanis, A. Savas-Halilaj, Th. Vlachos, “Complete minimal hypersurfaces in $\mathbb S^4$ with zero Gauss–Kronecker curvature”, Math. Proc. Cambridge Philos. Soc., 142:1 (2007), 125–132 | DOI | MR | Zbl

[8] A. C. Asperti, R. M. B. Chaves, L. A. M. Sousa Jr., “The Gauss–Kronecker curvature of minimal hypersurfaces in four-dimensional space forms”, Math. Z., 267:3-4 (2011), 523–533 | DOI | MR | Zbl

[9] T. Lusala, M. Scherfner, L. A. M. Sousa Jr., “Closed minimal Willmore hypersurfaces of $\mathbb S^5(1)$ with constant scalar curvature”, Asian J. Math., 9:1 (2005), 65–78 | DOI | MR | Zbl

[10] M. Dajczer, D. Gromoll, “Gauss parametrizations and rigidity aspects of submanifolds”, J. Differential Geom., 22:1 (1985), 1–12 | DOI | MR | Zbl

[11] M. P. do Carmo, O método do referencial móvel, Instituto de Matemática Pura e Aplicada, Rio de Janeiro, 1976, vi+231 pp.

[12] R. S. Palais, Chuu-lian Terng, Critical point theory and submanifold geometry, Lecture Notes in Math., 1353, Springer-Verlag, Berlin, 1988, x+272 pp. | DOI | MR | Zbl

[13] D. Ferus, “On the completeness of nullity foliations”, Michigan Math. J., 18:1 (1971), 61–64 | DOI | MR | Zbl

[14] E. L. Lima, Fundamental groups and covering spaces, Transl. from the Portuguese, A K Peters, Ltd., Natick, MA, 2003, x+210 pp. | MR | Zbl

[15] I. Tamura, Topologiya sloenii, Mir, M., 1979, 319 pp. ; I. Tamura, Yōsō no toporojī, Sūgaku Sensho, Iwanami Shoten, Tokyo, 1976, 238 с. ; Topology of foliations: an introduction, Transl. Math. Monogr., 97, Amer. Math. Soc., Providence, RI, 1992, xii+193 с. | MR | Zbl | MR | Zbl | MR | Zbl

[16] R. S. Palais, A global formulation of the Lie theory of transformation groups, Mem. Amer. Math. Soc., no. 22, Amer. Math. Soc., Providence, R.I., 1957, iii+123 pp. | MR | Zbl