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)



Codimension one isometric immersions between Lorentz spaces

Author: L. K. Graves
Journal: Trans. Amer. Math. Soc. 252 (1979), 367-392
MSC: Primary 53C50; Secondary 53C42
MathSciNet review: 534127
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The theorem of Hartman and Nirenberg classifies codimension one isometric immersions between Euclidean spaces as cylinders over plane curves. Corresponding results are given here for Lorentz spaces, which are Euclidean spaces with one negative-definite direction (also known as Minkowski spaces). The pivotal result involves the completeness of the relative nullity foliation of such an immersion. When this foliation carries a nondegenerate metric, results analogous to the Hartman-Nirenberg theorem obtain. Otherwise, a new description, based on particular surfaces in the three-dimensional Lorentz space, is required.

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

  • [A] Kinetsu Abe, A characterization of totally geodesic submanifolds in 𝑆^{𝑁} and 𝐶𝑃^{𝑁} by an inequality, Tôhoku Math. J. (2) 23 (1971), 219–244. MR 0295253
  • [B] W. B. Bonnor, Null curves in a Minkowski space-time, Tensor (N.S.) 20 (1969), 229–242. MR 0248658
  • [F] Dirk Ferus, On the completeness of nullity foliations, Michigan Math. J. 18 (1971), 61–64. MR 0279733
  • [G] W. Greub, Linear algebra, 2nd ed., Springer-Verlag, New York, 1963.
  • [HN] Philip Hartman and Louis Nirenberg, On spherical image maps whose Jacobians do not change sign, Amer. J. Math. 81 (1959), 901–920. MR 0126812
  • [KN] Shoshichi Kobayashi and Katsumi Nomizu, Foundations of differential geometry. Vol I, Interscience Publishers, a division of John Wiley & Sons, New York-London, 1963. MR 0152974
  • [M] John Douglas Moore, Isometric immersions of riemannian products, J. Differential Geometry 5 (1971), 159–168. MR 0307128
  • [N$ _{1}$] Katsumi Nomizu, On hypersurfaces satisfying a certain condition on the curvature tensor, Tôhoku Math. J. (2) 20 (1968), 46–59. MR 0226549
  • [N$ _{2}$] -, Lectures on differential geometry of submanifolds (unpublished lecture notes), Brown University, 1975.
  • [S] M. Spivak, A comprehensive introduction to differential geometry. II, III, Publish or Perish, Boston, Mass., 1970, 1975.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 53C50, 53C42

Retrieve articles in all journals with MSC: 53C50, 53C42

Additional Information

Keywords: Lorentz spaces, isometric immersions, null curves, null frames, relative nullity foliation, (non)degenerate relative nullities, complete relative nullities, Hartman-Niren-berg theorem, cylinders over plane curves, B-scrolls
Article copyright: © Copyright 1979 American Mathematical Society