On a variational formula for the H. Weyl curvature invariants
We use the ring of curvature structures, to answer a question of M. Berger about a variational formula for the H. Weyl curvature invariants. Recall that the (2k)-H. Weyl curvature invariant is a polynomial of degree 2k with respect to the Riemannian curvature. In dimension 2k it is just the Gauss-Bonnet integrand, in higher dimensions it appears naturally, as an integrand, in the well known H. Weyl’s tube formula. Mathematics Subject Classification (2000). 58E11, 58C99.
Double forms, curvature structures and the $(p,q)$-curvatures
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 > 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.
Representations and computations of {2, 3~} and {2, 4~}-inverses in indefinite inner product spaces
We investigate full-rank representations of { 2 , 3 ~ } and { 2 , 4 ~ } -inverses with prescribed rank as well as with prescribed range and null space in an indefinite inner product space. In certain particular cases we obtain new results or generalizations related with { 2 , 3 } and { 2 , 4 } -inverses in a Hilbert (unitary) space. Direct and iterative methods for computing { 2 , 3 ~ } and { 2 , 4 ~ } -inverses in an indefinite inner product space are developed. Numerical examples with different metric matrices are presented.