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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Factorization of curvature operators
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by Jaak Vilms PDF
Trans. Amer. Math. Soc. 260 (1980), 595-605 Request permission


Let V be a real finite-dimensional vector space with inner product and let R be a curvature operator, i.e., a symmetric linear map of the bivector space $\Lambda { ^2}V$ into itself. Necessary and sufficient conditions are given for R to admit factorization as $R = \Lambda { ^2}L$, with L a symmetric linear map of V into itself. This yields a new characterization of Riemannian manifolds that admit local isometric embedding as hypersurfaces of Euclidean space.
    E. Bertini, Introduzione alla geometria proiettiva degli iperspazi, 2nd ed., Principato, Messina, 1923.
  • Wei-Liang Chow, On the geometry of algebraic homogeneous spaces, Ann. of Math. (2) 50 (1949), 32–67. MR 28057, DOI 10.2307/1969351
  • Jean Dieudonné, La géométrie des groupes classiques, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1963 (French). Seconde édition, revue et corrigée. MR 0158011
  • Howard Jacobowitz, Curvature operators on the exterior algebra, Linear and Multilinear Algebra 7 (1979), no. 2, 93–105. MR 529876, DOI 10.1080/03081087908817264
  • Shoshichi Kobayashi and Katsumi Nomizu, Foundations of differential geometry. Vol. II, Interscience Tracts in Pure and Applied Mathematics, No. 15 Vol. II, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1969. MR 0238225
  • Marvin Marcus, Finite dimensional multilinear algebra. Part II, Pure and Applied Mathematics, Vol. 23, Marcel Dekker, Inc., New York, 1975. MR 0401796
  • N. A. Rozenson, On Riemannian spaces of class one, Izv. Akad. Nauk SSSR Ser. Mat. 4 (1940), 181-192; 5 (1941), 325-351; 7 (1943), 253-284.
  • T. Y. Thomas, Riemann spaces of class one and their characterization, Acta Math. 67 (1936), no. 1, 169–211. MR 1555419, DOI 10.1007/BF02401741
  • Jaak Vilms, Local isometric imbedding of Riemannian $n$-manifolds into Euclidean $(n+1)$-space, J. Differential Geometry 12 (1977), no. 2, 197–202. MR 487854
  • Hassler Whitney, Geometric integration theory, Princeton University Press, Princeton, N. J., 1957. MR 0087148
  • N. N. Yanenko, Some questions of the theory of imbedding of Riemannian metrics in Euclidean spaces, Uspehi Matem. Nauk (N.S.) 8 (1953), no. 1(53), 21–100 (Russian). MR 0055758
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Additional Information
  • © Copyright 1980 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 260 (1980), 595-605
  • MSC: Primary 53C20; Secondary 15A63, 53B25, 53C40
  • DOI:
  • MathSciNet review: 574802