References
Aleksandrov, A.D.: Uniqueness theorems for surfaces in the large. I–V. Vestnik. Leningrad. Univ.11, 5–17 (1956);12, 15–44 (1957);13, 14–26 (1958);13, 27–34 (1958);13, 5–8 (1958). Amer. Math. Soc. Transl. (2)21, 341–416 (1962)
Alexander, S.: Locally convex hypersurfaces of negatively curved spaces. Proc. Am. Math. Soc.64, 321–326 (1977)
Backes, E., Reckziegel, H.: On symmetric submanifolds of spaces of constant curvature. Math. Ann.263, 419–433 (1983)
Bishop, R.L.: Infinitesimal convexity implies local convexity. Indiana Univ. Math. J.24, 169–172 (1974)
Bishop, R.L., O'Neill, B.: Manifolds of negative curvature. Trans. Am. Math. Soc.145, 1–49 (1969)
Cartan, E.: Familles de surfaces isoparamétrique dans les espaces à courbure constante. Ann. Mat. Pura Appl.17, 177–191 (1938)
Cheng, S.Y., Yau, S.T.: Hypersurfaces with constant scalar curvature. Math. Ann.225, 195–204 (1977)
Delaunay, C.: Sur la surface de révolution dont la courbure moyenne est constante. J. Math. Pures Appl.6, 309–320 (1841)
Do Carmo, M.P., Warner, F.W.: Rigidity and convexity of hypersurfaces in spheres. J. Differential Geometry4, 133–144 (1970)
Ferus, D.: Symmetric submanifolds of euclidean space. Math. Ann.247, 81–93 (1980)
Greene, R.E., Wu, H.:C ∞-convex functions and manifolds of positive curvature. Acta Math.137, 209–245 (1976)
Hartman, P.: On complete hypersurfaces of nonnegative sectional curvatures and constantm-th curvature. Trans. Am. Math. Soc.245, 363–374 (1978)
Hsiang, W.Y.: On generalization of theorems of A. D. Alexandrov and C. Delaunay on hypersurfaces of constant mean curvature. Duke Math. J.49, 485–496 (1982)
Hsiang, W.Y.: Generalized rotational hypersurfaces of constant mean curvature in the Euclidean spaces. I. J. Differential Geometry17, 337–356 (1982)
Huck, H., Roitzsch, R., Simon, U., Vortisch, W., Walden, R., Wegner, B., Wendland, W.: Beweismethoden der Differentialgeometrie im Großen. Lecture Notes in Mathematics, Vol. 335. Berlin, Heidelberg, New York: Springer 1973
Klingenberg, W.: Riemannian geometry. Berlin, New York: De Gruyter 1982
Kobayashi, S., Nomizu, K.: Foundations of differential geometry. Vol. I/II. New York, London, Sydney: Interscience 1963/69
Lawson, H.B., Jr.: Local rigidity theorems for minimal hypersurfaces. Ann. Math.89, 187–197 (1969)
Marcus, M., Lopes, L.: Inequalities for symmetric functions and Hermitian matrices. Canad. J. Math.8, 524–531 (1956)
Münzner, H.F.: Weylsche Ungleichungen für Hyperflüchen imE n+1. Lecture, held on the Geometrie-Tagung 1971 in Oberwolfach (and handwritten letter to U. Simon)
Nomizu, K., Smyth, B.: A formula of Simon's type and hypersurfaces with constant mean curvature. J. Differential Geometry3, 367–377 (1969)
Reckziegel, H.: On the problem whether the image of a given differentiable map into a Riemannian manifold is contained in a submanifold with parallel second fundamental form. J. Reine Angew. Math.325, 87–104 (1981)
Reckziegel, H.: A class of distinguished isometric immersions with parallel second fundamental form. Result. Math.6, 56–63 (1983)
Reilly, R.C.: Extrinsic rigidity theorems for compact submanifolds of the sphere. J. Differential Geometry4, 487–497 (1970)
Reilly, R.C.: Variational properties of functions of the mean curvatures for hypersurfaces in space forms. J. Differential Geometry8, 465–477 (1973)
Reilly, R.C.: Applications of the Hessian operator in a Riemannian manifold. Indiana Univ. Math. J.26, 459–472 (1977)
Sacksteder, R.: On hypersurfaces with no negative sectional curvatures. Am. J. Math.82, 609–630 (1960)
Simon, U.: A further method in global differential geometry. Abh. Math. Sem. Univ. Hamburg44, 52–69 (1975)
Simon, U.: Codazzi tensors. In: Global differential geometry and global analysis. Proc. Berlin 1979, eds. D. Ferus, W. Kühnel, U. Simon, B. Wegner. Lecture Notes in Mathematics, Vol. 838, p. 289–296. Berlin, Heidelberg, New York: Springer 1981
Singley, D.H.: Integral formulas for pairs of metrics with applications to convex hypersurfaces. (preprint, based on the author's Ph. D. Thesis, Columbia Univ. (1971)
Spivak, M.: A comprehensive introduction to differential geometry. IV. Boston: Publish or Perish 1975
Strübing, W.: Symmetric submanifolds of Riemannian manifolds. Math. Ann.245, 37–44 (1979)
Waerden, B.L. Van der: Algebra I. Grundlehren d. math. Wiss. Bd. 33. Berlin, Göttingen, Heidelberg: Springer 1955
Voss, K.: Einige differentialgeometrische Kongruenzsätze für geschlossene Flächen und Hyperflächen. Math. Ann.131, 180–218 (1956)
Walter, R.: Differentialgeometrie. Mannheim, Wien, Zürich: Bibliographisches Institut/Wissenschaftsverlag 1978
Walter, R.: Konvexität in riemannschen Mannigfaltigkeiten. Jber. DMV83, 1–31 (1981)
Weyl, H.: Über die Bestimmung einer geschlossenen konvexen Fläche durch ihr Linienelement. Vierteljahresschrift. Naturforsch. Ges. Zürich61, 40–72 (1916)
Wolf, J.: Spaces of constant curvature. New York: McGraw Hill 1967
Wu, H.: The Bochner technique. Differential geometry and differential equations. Proc. 1980 Beijing Sympos.2, 929–1071 (1982)
Yano, K., Bochner, S.: Curvature and Betti Numbers. Ann. of Math. Studies 32. Princeton: Princeton Univ. Press 1953
Yau, S.T.: Submanifolds with constant mean curvature, I, II. Am. J. Math.96, 346–366 (1974); ibid. Yau, S.T.: Submanifolds with constant mean curvature, I, II. Am. J. Math.97, 76–100 (1975)
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Walter, R. Compact hypersurfaces with a constant higher mean curvature function. Math. Ann. 270, 125–145 (1985). https://doi.org/10.1007/BF01455537
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DOI: https://doi.org/10.1007/BF01455537