Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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

 
 

 

Factorization of curvature operators


Author: Jaak Vilms
Journal: Trans. Amer. Math. Soc. 260 (1980), 595-605
MSC: Primary 53C20; Secondary 15A63, 53B25, 53C40
DOI: https://doi.org/10.1090/S0002-9947-1980-0574802-7
MathSciNet review: 574802
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: 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.


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

  • [1] E. Bertini, Introduzione alla geometria proiettiva degli iperspazi, 2nd ed., Principato, Messina, 1923.
  • [2] W. L. Chow, On the geometry of algebraic homogeneous spaces, Ann. of Math. (2) 50 (1949), 32-67. MR 0028057 (10:396d)
  • [3] J. Dieudonné, La géométrie des groupes classiques, 2nd ed., Springer-Verlag, Berlin, 1963. MR 0158011 (28:1239)
  • [4] H. Jacobowitz, Curvature operators on the exterior algebra, Linear and Multilinear Algebra 7 (1979), 93-105. MR 529876 (80c:53048)
  • [5] S. Kobayashi and K. Nomizu, Foundations of differential geometry, Volume II, Interscience, New York, 1969. MR 0238225 (38:6501)
  • [6] M. Marcus, Finite-dimensional multilinear algebra, Vols. I and II, Dekker, New York, 1974 and 1975. MR 0401796 (53:5623)
  • [7] 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.
  • [8] T. Y. Thomas, Riemannian spaces of class one and their characterization, Acta Math. 67 (1936), 169-211. MR 1555419
  • [9] J. Vilms, Local isometric imbedding of Riemannian n-manifolds into Euclidean $ (n\, + \,1)$-space, J. Differential Geometry 12 (1977), 197-202. MR 0487854 (58:7452)
  • [10] H. Whitney, Geometric integration theory, Princeton Univ. Press, Princeton, N. J., 1957. MR 0087148 (19:309c)
  • [11] N. N. Yanenko, Some questions of the theory of imbeddings of Riemannian metrics into Euclidean spaces, Uspehi Mat. Nauk (N.S.) 8, no. 1 (53), (1953), 21-100. MR 0055758 (14:1122f)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 53C20, 15A63, 53B25, 53C40

Retrieve articles in all journals with MSC: 53C20, 15A63, 53B25, 53C40


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1980-0574802-7
Article copyright: © Copyright 1980 American Mathematical Society

American Mathematical Society