zbMATH — the first resource for mathematics

Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Hypersurfaces with null higher order mean curvature. (English) Zbl 1210.53057
The constancy condition of higher elementary symmetric functions of the principal curvatures (= higher mean curvatures ${H}_{r}$) is an interesting problem in the global theory of hypersurfaces. For the compact case and a non-vanishing constant see [R. Walter, Math. Ann. 270, 125–145 (1985; Zbl 0536.53054)]. In the present paper, the authors deal with the case of vanishing higher mean curvature of complete and orientable hypersurfaces of dimension $n$ in space forms. They make the assumption that an open part does not consist of totally geodesic hypersurfaces. Another assumption is that ${H}_{r+1}$ vanishes identically and ${H}_{r}$ never vanishes. One of the results states that, under these assumptions, the manifold is foliated by complete totally geodesic submanifolds of dimension $n-r$.
MSC:
 53C42 Immersions (differential geometry) 53A10 Minimal surfaces, surfaces with prescribed mean curvature
References:
 [1] H. Alencar. Hipersuperfícies Mínimas de $ℝ$2m Invariantes por SO(m) $×$ SO(m). Doctor Thesis, IMPA – Brazil (1988). [2] H. Alencar and A.G. Colares. Integral Formulas for the r-Mean Curvature Linearized Operator of a Hypersurface. Annals of Global Analysis and Geometry, 16 (1998), 203–220. · Zbl 0920.53030 · doi:10.1023/A:1006555603714 [3] H. Alencar, M. do Carmo and M.F. Elbert. Stability of Hypersurface with Vanishing r-Mean Curvatures in Euclidean spaces. J. Reine Angew. Math., 554 (2003), 201–216. · Zbl 1093.53063 · doi:10.1515/crll.2003.006 [4] H. Alencar and K. Frensel. Hypersurface Whose Tangent Geodesic Omit a Nonempty Set. Differential Geometry – A Symposium In Honour of Manfredo do Carmo, ed. New York: Longman Scientific & Technical (1991), 1–13. [5] A. Caminha. On Hypersurface into Riemannian Space of Constant Sectional Curvature. Kodai Math J., 29 (2006), 185–210. · Zbl 1107.53037 · doi:10.2996/kmj/1151936435 [6] A. Caminha. Complete Spacelike Hypersurfaces in Conformally Stationary Lorentz manifolds. Gen. Relativ Gravit, 41 (2009), 173–189. · Zbl 1162.83304 · doi:10.1007/s10714-008-0663-z [7] M. Dajczer et al. Submanifolds and Isometric Immersions. Publish or Perish, Houston (1990). [8] M. Dacjzer and D. Gromoll. Gauss Parametrizations and Rigidity Aspects of Submanifolds. J. Differential Geometry, 22 (1985), 1–12. [9] M. Dacjzer and D. Gromoll. On Spherical Submanifolds with Nullity. Proc. Am. Math. Soc., 93 (1985), 99–100. · doi:10.1090/S0002-9939-1985-0766536-0 [10] M.F. Elbert. Constant Positive 2-Mean Curvature Hypersurfaces. Illinois J. Math., 46(1) (2002), 247–267. [11] T. Hasanis and D. Koutroufiots. A Property of Complete Minimal Surfaces. Trans. Amer. Math. Soc., 281 (1984), 833–843. · doi:10.1090/S0002-9947-1984-0722778-5 [12] H. Rosenberg. Hypersurfaces of Constant Curvature in Space Forms. Bull. Sc. Math., 117 (1993), 217–239. [13] J. Sato. Stability of O(p + 1) $×$ O(p + 1)-Invariant Hypersurfaces with Zero Scalar Curvature in Euclidean Space. Annals of Global Analysis and Geometry, 22 (2002), 135–153. · Zbl 1036.53039 · doi:10.1023/A:1019536730847