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)



Double forms, curvature structures and the $(p,q)$-curvatures

Author: M.-L. Labbi
Journal: Trans. Amer. Math. Soc. 357 (2005), 3971-3992
MSC (2000): Primary 53B20, 53C21; Secondary 15A69
Published electronically: May 20, 2005
MathSciNet review: 2159696
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We introduce a natural extension of the metric tensor and the Hodge star operator to the algebra of double forms to study some aspects of the structure of this algebra. These properties are then used to study new Riemannian curvature invariants, called the $(p,q)$-curvatures. They are a generalization of the $p$-curvature obtained by substituting the Gauss-Kronecker tensor to the Riemann curvature tensor. In particular, for $p=0$, the $(0,q)$-curvatures coincide with the H. Weyl curvature invariants, for $p=1$ the $(1,q)$-curvatures are the curvatures of generalized Einstein tensors, and for $q=1$ the $(p,1)$-curvatures coincide with the $p$-curvatures.

Also, we prove that the second H. Weyl curvature invariant is nonnegative for an Einstein manifold of dimension $n\geq 4$, and it is nonpositive for a conformally flat manifold with zero scalar curvature. A similar result is proved for the higher H. Weyl curvature invariants.

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

  • 1. Avez, A., Applications de la formule de Gauss-Bonnet-Chern aux variétés de dimension 4. CRAS 256, 5488-5490 (1963). MR 0157320 (28:555)
  • 2. Besse, A. L., Einstein Manifolds, Springer-Verlag (1987). MR 0867684 (88f:53087)
  • 3. Kulkarni, R. S., On Bianchi Identities, Math. Ann. 199, 175-204 (1972). MR 0339004 (49:3767)
  • 4. Labbi, M. L. Variétés Riemanniennes à $p$-courbure positive, Thèse, Publication Université Montpellier II (1995), France.
  • 5. Labbi, M. L. On a variational formula for the H. Weyl curvature invariants, to appear.
  • 6. Chen, B.-Y., Dillen, F., Verstraelen, L., and Vrancken, L., Characterizations of Riemannian space forms, Einstein spaces and conformally flat spaces, Proceedings of the American Mathematical Society, volume 128, Number 2, pages 589-598 (1999). MR 1664333 (2000c:53050)
  • 7. Thorpe, J. A., Some remarks on the Gauss-Bonnet integral, Journal of Mathematics and Mechanics, Vol. 18, No. 8 (1969). MR 0256307 (41:963)
  • 8. Weyl, H., On the volume of tubes, Amer. J. Math., vol. 61, 461-472 (1939).

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 53B20, 53C21, 15A69

Retrieve articles in all journals with MSC (2000): 53B20, 53C21, 15A69

Additional Information

M.-L. Labbi
Affiliation: Department of Mathematics, College of Science, University of Bahrain, Isa Town 32038, Bahrain

Keywords: Double form, curvature structure, $(p,q)$-curvature, Gauss-Kronecker curvature, H. Weyl curvature invariants
Received by editor(s): July 22, 2003
Published electronically: May 20, 2005
Article copyright: © Copyright 2005 American Mathematical Society

American Mathematical Society