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