Abstract
We investigate (0, 2)-tensors, which fulfil Codazzi-equations, on closed Riemannian manifolds with nonnegative sectional curvature, and give various applications in global differential geometry.
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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. Springer Berlin etc. 1970.
L. P. Eisenhart, Riemannian geometry. Fourth print. Princeton Univ. Press 1960.
P. Franzke, Minimale Untermannigfaltigkeiten von Sphären. Math. Nachrichten56 (1973) 155–159.
E. Glodek, Some remarks on conformally symmetric spaces. Colloquium Math.22 (1971) 121–123.
D. A. Hoffmann, Surfaces in constant curvature manifolds with parallel mean curvature vector field. Bull. Amer. Math. Soc.78 (1972) 247–250.
H. Huck, R. Roitzsch, U. Simon, W. Vortisch, R. Walden, B. Wegner, W. Wendland, Beweismethoden der Differentialgeometrie im Großen. Lecture Notes in Mathematics 335. Springer Berlin etc. 1973.
S. Kobayashi, K. Nomizu, Foundations of Differential Geometry I, II. Interscience Publ. New York 1963, 1969.
A. Lichnérowicz, Opérateurs différentiels invariant sur un espace homogène. Ann. Sci. Ec. Norm. Sup. 3,81 (1965) 341–385.
H. F. Münzner, Weylsche Ungleichungen. manuscript. Universität Bremen 1972.
H. Nakagawa, I. Yokote, On hypersurfaces with constant scalar curvature in a Riemannian manifold of constant curvature. Kòdai Math. Sem. Rep.24 (1972) 471–481.
P. J. Ryan, A note on conformally flat spaces with constant scalar curvature. Proc. 13th biennal Sem. Canadian Math. Congress. Vol. 2 (1972) 115–124.
U. Simon, Minkowskische Integralformeln und ihre Anwendungen in der Differentialgeometrie im Großen. Math. Ann.173 (1967) 307–321.
U. Simon, Probleme der lokalen und globalen mehrdimensionalen Differentialgeometrie. manuscripta math.2 (1970) 241–284.
U. Simon, Compact conformally symmetric Riemannian spaces. Math. Zeitschrift132 (1973) 173–177.
U. Simon, On differential operators of second order on Riemannian manifolds with nonpositive curvature. Colloquium Math.31 (1974) 223–229.
U. Simon, A. Weinstein, Anwendungen der de Rhamschen Zerlegung auf Probleme der lokalen Flächentheorie. manuscripta math.1 (1969) 139–146.
J. Simons, Minimal varieties in Riemannian manifolds. Annals of Math.88 (1968) 62–105.
B. Smyth, Submanifolds of constant mean curvature. Math. Ann.205 (1973) 265–280.
T. Sumitomo, On a certain class of Riemannian homogeneous spaces. Colloquium Math.26 (1972) 129–133.
T. Sumitomo, On the commutator of differential operators. Hokkaido Math. J.1 (1972) 30–42.
K. Voss, Isometrie von Flächen bezüglich der zweiten Fundamentalform. Nachr. Österr. Math. Ges.91 (1970) 73.
B. Wegner, Codazzi-tensoren und Kennzeichnungen sphärischer Immersionnen. J. Diff. Geom.9 (1974) 61–70.
B. Wegner, Kennzeichnungen von Räumen konstanter Krümmung unter lokal konform euklidischen Riemannschen Mannigfaltigkeiten. Geometriae Dedicata2 (1973) 269–281.
B. Wegner, Eine Charakterisierung von Produkten geschlossener sphärischer Kurven. Math. Nachr.59 (1974) 229–234.
K. Yano, B. Y. Chen, Minimal submanifolds of a higher dimensional sphere. Tensor22 (1971) 369–373.
D. Singley, Pairs of metrics on Riemannian manifolds. Ph. D. Thesis. Columbia University 1971.
K. P. Grotemèyer, Ein Satz aus der Differentialgeometrie der Eiflächen. Archiv Math.6 (1955) 403–407.
K. Nomizu andB. Smyth, A formula of Simon's type and hypersurfaces with constant mean curvature. J. Diff. Geom.3 (1969) 367–377.
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Dedicated toE. Sperner on his seventieth birthday
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Simon, U. A further method in global differential geometry. Abh.Math.Semin.Univ.Hambg. 44, 52–69 (1975). https://doi.org/10.1007/BF02992946
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DOI: https://doi.org/10.1007/BF02992946