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A holonomy proof of the positive curvature operator theorem


Author: W. A. Poor
Journal: Proc. Amer. Math. Soc. 79 (1980), 454-456
MSC: Primary 53C20
MathSciNet review: 567991
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Abstract: Extending work of Bochner-Yano and M. Berger, D. Meyer proved that if the curvature operator of a compact, oriented, Riemannian manifold M has positive eigenvalues, then M is a rational homology sphere. Here a proof is given using Chern's holonomy formula for the Laplacian on M; for completeness, a quick proof of Chern's formula is included.


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  • [1] Marcel Berger, Sur les variétés à opérateur de courbure positif, C. R. Acad. Sci. Paris 253 (1961), 2832–2834 (French). MR 0140055
  • [2] S. Bochner and K. Yano, Tensor-fields in non-symmetric connections, Ann. of Math. (2) 56 (1952), 504–519. MR 0054325
  • [3] Shiing-shen Chern, On a generalization of Kähler geometry, Algebraic geometry and topology. A symposium in honor of S. Lefschetz, Princeton University Press, Princeton, N. J., 1957, pp. 103–121. MR 0087172
  • [4] S. Gallot and D. Meyer, Opérateur de courbure et laplacien des formes différentielles d’une variété riemannienne, J. Math. Pures Appl. (9) 54 (1975), no. 3, 259–284 (French). MR 0454884
  • [5] Henry Maillot, Sur les variétés riemanniennes à opérateur de courbure pur, C. R. Acad. Sci. Paris Sér. A 278 (1974), 1127–1130. MR 0400109
  • [6] Daniel Meyer, Sur les variétés riemanniennes à opérateur de courbure positif, C. R. Acad. Sci. Paris Sér. A-B 272 (1971), A482–A485 (French). MR 0279736
  • [7] K. Yano and S. Bochner, Curvature and Betti numbers, Annals of Mathematics Studies, No. 32, Princeton University Press, Princeton, N. J., 1953. MR 0062505
  • [8] A. Weil, Un théorème fondamental de Chern en géométrie riemannienne, Séminaire Bourbaki, 14e année, 1961/62, no. 239.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1980-0567991-7
Keywords: Eigenvalue of curvature operator, Laplacian, rational homology sphere, holonomy algebra
Article copyright: © Copyright 1980 American Mathematical Society