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A normal form for a special class of curvature operators


Author: Stanley M. Zoltek
Journal: Proc. Amer. Math. Soc. 79 (1980), 614-618
MSC: Primary 53B20; Secondary 15A75
DOI: https://doi.org/10.1090/S0002-9939-1980-0572314-3
MathSciNet review: 572314
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Abstract: In the case of a 4-dimensional oriented inner product space Singer and Thorpe found a canonical form for a curvature operator which commutes with a generator of $ {\Lambda ^4}$, and used it to prove that the curvature function is completely determined by its critical point behavior. In dimension 5 we extend these results to curvature operators which commute with an element of $ {\Lambda ^4}$.


References [Enhancements On Off] (What's this?)

  • [1] S. Kobayashi and K. Nomizu, Foundations of differential geometry, Vols. I, II, Interscience, New York, 1969. MR 38 #6501. MR 0152974 (27:2945)
  • [2] I. M. Singer and J. A. Thorpe, The curvature of 4-dimensional Einstein spaces, Global Analysis, Papers in Honor of K. Kodaira, Univ. of Tokyo Press, Tokyo, 1969, pp. 355-365. MR 41 #959. MR 0256303 (41:959)
  • [3] John A. Thorpe, The zeroes of nonnegative curvature operators, J. Differential Geometry 5 (1971), 113-125. MR 44 #7469; Erratum, 11 (1976). MR 0290285 (44:7469)
  • [4] Stanley M. Zoltek, Non-negative curvature operators: Some non-trivial examples, J. Differential Geometry (to appear). MR 587555 (81m:53037)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1980-0572314-3
Keywords: Curvature operator, normal form, critical point behavior, Einstein manifold, Bianchi identity, Grassmann quadratic 2-relations
Article copyright: © Copyright 1980 American Mathematical Society

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