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Kleinjohann, Norbert; Walter, Rolf

Nonnegativity of the curvature operator and isotropy for isometric immersions. (English) Zbl 0476.53032

Math. Z. 181, 129-142 (1982).
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Cited in 3 Documents

MSC:

53C40 Global submanifolds
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)

Keywords:

isometric immersion; nonnegative sectional curvature; isotropic immersion; Veronese surface; second fundamental form

Citations:

Zbl 0386.53031; Zbl 0171.205
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Full Text: DOI EuDML

References:

[1] Allendoerfer, C.B., Weil, A.: The Gauss-Bonnet theorem for riemannian polyhedra. Trans. Amer. Math. Soc.53, 101-129 (1943) · Zbl 0060.38102 · doi:10.1090/S0002-9947-1943-0007627-9
[2] Bangert, V.: Riemannsche Mannigfaltigkeiten mit nicht-konstanter konvexer Funktion. Arch. Math. (Basel)31, 163-170 (1978) · Zbl 0372.53021
[3] Bangert, V.: Totally convex sets in complete Riemannian manifolds. J. Differential Geometry16, 333-345 (1981) · Zbl 0479.53032
[4] Bourguignon, J.P., Karcher, H.: Curvature operators: Pinching estimates and geometric examples. Ann. Sci. École Norm. Sup. Sér. 411, 71-92 (1978) · Zbl 0386.53031
[5] Cheeger, J., Gromoll, D.: On the structure of complete manifolds of nonnegative curvature. Ann. of Math. (2)96, 413-443 (1972) · Zbl 0246.53049 · doi:10.2307/1970819
[6] Chern, S.S.: A simple intrinsic proof of the Gauss-Bonnet formula for closed riemannian manifolds. Ann. of Math. (2)45, 747-752 (1944) · Zbl 0060.38103 · doi:10.2307/1969302
[7] Chern, S.S.: On curvature and characteristic classes of a riemannian manifold. Abh. Math. Sem. Univ. Hamburg20, 117-126 (1955) · Zbl 0066.17003 · doi:10.1007/BF02960745
[8] Chern, S.S.: On minimal spheres in the four-sphere. In: Studies and essays (presented to W. W. Chen on his sixtieth birthday) (Taipei 1970), pp. 137-150. Taipei: Mathematical Research Center 1970
[9] Chern, S.S., Do Carmo, M., Kobayashi, S.: Minimal submanifolds of a sphere with second fundamental form of constant length. In: Functional Analysis and Related Fields. Proceedings of a Conference in honor of Professor Marshall Stone (Chicago 1968), pp. 59-75. Berlin-Heidelberg-New York: Springer 1970 · Zbl 0216.44001
[10] Cohn-Vossen, St.: Kürzeste Wege und Totalkrümmung auf Flächen. Compositio Math.2, 69-133 (1935) · JFM 61.0789.01
[11] Geroch, R.: Positive sectional curvature does not imply positive Gauss-Bonnet integrand. Proc. Amer. Math. Soc.54, 267-270 (1976) · Zbl 0325.53042 · doi:10.1090/S0002-9939-1976-0390961-8
[12] Greene, R.E., Wu, H.: Integrals of subharmonic functions on manifolds of nonnegative curvature. Invent Math.27, 265-298 (1974) · Zbl 0342.31003 · doi:10.1007/BF01425500
[13] Greene, R.E., Wu, H.: Approximation theorems,C ? convex exhaustions and manifolds of positive curvature. Bull. Amer. Math. Soc.81, 101-104 (1975) · Zbl 0307.53026 · doi:10.1090/S0002-9904-1975-13653-6
[14] Greene, R.E., Wu, H.:C ? convex functions and manifolds of positive curvature. Acta Math.137, 209-245 (1976) · Zbl 0372.53019 · doi:10.1007/BF02392418
[15] Greub, W.: Lineare Algebra, korr. Nachdr. d. 1. Aufl. Berlin-Heidelberg-New York: Springer 1976 · Zbl 0325.15003
[16] Itoh, T., Ogiue, K.: Isotropic immersions and Veronese manifolds. Trans. Amer. Math. Soc.209, 109-117 (1975) · Zbl 0315.53029 · doi:10.1090/S0002-9947-1975-0375172-8
[17] Jacobowitz, H.: Curvature operators on the Exterior Algebra, Linear and Multilinear Algebra7, 93-105 (1979) · Zbl 0402.53026 · doi:10.1080/03081087908817264
[18] Klembeck, P.: On Geroch’s counterexample to the algebraic Hopf conjecture. Proc. Amer. Math. Soc.59, 334-336 (1976) · Zbl 0341.53011
[19] Kulkarni, R.S.: On the Bianchi identities. Math. Ann.199, 175-204 (1972) · doi:10.1007/BF01429873
[20] Moore, J.D.: An application of second variation to submanifold theory. Duke Math. J.42, 191-193 (1975) · Zbl 0337.53045 · doi:10.1215/S0012-7094-75-04217-9
[21] Nakagawa, H., Itoh, T.: On isotropic immersions of space forms into a space form. In: Minimal Submanifolds and Geodesics. Proceeding of a Seminar (Tokyo 1977), Amsterdam-New York-Oxford: North Holland 1979 · Zbl 0434.53043
[22] O’Neill, B.: Isotropic and Kähler immersions. Canad. J. Math.17, 907-915 (1965) · Zbl 0171.20503 · doi:10.4153/CJM-1965-086-7
[23] Otsuki, T.: On the existence of solutions of a system of quadratic equations and its geometrical application. Proc. Japan Acad.29, 99-100 (1953) · Zbl 0052.17602 · doi:10.3792/pja/1195570690
[24] Poor, W.A., Jr.: Some results on nonnegatively curved manifolds. J. Differential Geometry9, 583-600 (1974) · Zbl 0292.53037
[25] Santalo, L.A.: Convex regions on then-dimensional spherical surface. Ann. of Math. (2)47, 448-459 (1946) · Zbl 0061.38007 · doi:10.2307/1969084
[26] Walter, R.: On the metric projection onto convex sets in riemannian spaces. Arch. Math. (Basel)25, 91-98 (1974) · Zbl 0311.53054
[27] Walter, R.: A generalized Allendoerfer-Weil formula and an inequality of the Cohn-Vossen type. J. Differential Geometry10, 167-180 (1975) · Zbl 0308.53042
[28] Walter, R.: Some analytical properties of geodesically convex sets. Abh. Math. Sem. Univ. Hamburg45, 263-282 (1976) · Zbl 0332.53026 · doi:10.1007/BF02992922
[29] Walter, R.: Differentialgeometrie. Mannheim-Wien-Zürich: Bibliographisches Institut-Wissenschaftsverlag 1978
[30] Walter, R.: Konvexität in riemannschen Mannigfaltigkeiten. Jber. Deutsch. Math.-Verein.83, 1-31 (1981) · Zbl 0469.53034
[31] Weinstein, A.: Positively curvedn-manifolds in ? n+1, J. Differential Geometry4, 1-4 (1970) · Zbl 0194.52903
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