Double forms, curvature structures and the \((p,q)\)-curvatures. (English) Zbl 1077.53033

Labbi, M.-L.

Trans. Am. Math. Soc. 357, No. 10, 3971-3992 (2005).

Summary: 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 by the Riemann curvature tensor. In particular, for \(p=0\), the \((0,q)\)-curvatures coincide with H. Weyl’s 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 proven for higher H. Weyl curvature invariants.
Given an \(n\)-dimensional Riemannian manifold \((M,g)\), let \(\Lambda(M)=\bigoplus_{p\geq 0}\Lambda^p(M)\) be the ring of differentiable forms on \(M\) and denote by \(D=\bigoplus_{p,q\geq 0}D^{p,q}\), where \(D^{p,q}=\Lambda^p(M)\otimes\Lambda^q(M)\), the ring of double forms on \(M\). The author defines a natural inner product \(\langle\;,\;\rangle\) on \(D\) and proves that the contraction map \(c\) is the adjoint of the multiplication map by \(g\). A natural extension of the Hodge operator \(*\) to \(D\) is defined and a formula relating the multiplication by \(g\) to the map \(c\) and the operator \(*\) is derived. This allows to define a canonical orthogonal decomposition of \(D^{p,q}\) and to give explicit formulas for the projections onto each factor. In particular, the author reobtains the well known decomposition of the Riemannian curvature tensor \(R\in D^{2,2}\).
Further formulas involving curvature structures which satisfy the first Bianchi identity are stated. We recall that a curvature structure is an element of the algebra \(C=\bigoplus_{p\geq 0} C^p,C^p\) consisting of the symmetric elements of \(D^{p,p}\). The author introduces new Riemannian curvature invariants \(R_{(p,q)}\) and their sectional curvatures \(s_{(p,q)}\). In particular, \(R_{(1,1)}\) is the Einstein tensor and, for any \(q\), \(s_{(0,q)}\) is a scalar function which coincides, up to a constant, with the Weyl curvature invariant \(h_{2q}\). Several examples and interesting results involving these invariants are given. We only mention the following theorem: Let \((M,g)\) be an \(n\)-dimensional Riemanman manifold, \(n\geq 4\). If \((M,g)\) is Einstein, then \(h_4\geq 0\) and \(h_4=0\) if and only if \(M\) is flat. If \((M,g)\) is conformally flat with zero scalar curvature, then \(h_4< 0\) and \(h_4=0\) if and only if \(M\) is flat.

Reviewer: Maria Falcitelli (Bari)

MSC:

53C20	Global Riemannian geometry, including pinching
53C21	Methods of global Riemannian geometry, including PDE methods; curvature restrictions
15A69	Multilinear algebra, tensor calculus
58A10	Differential forms in global analysis
58A14	Hodge theory in global analysis

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