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On the decomposition of global conformal invariants. I. (English) Zbl 1190.53028

Let \(P\) be a universal polynomial scalar invariant in the category of \(n\) dimensional Riemannian manifolds. H. Weyl’s theorem shows that such invariants arise as universal polynomials in the curvature tensor and its covariant derivatives via contractions of indices. Singer proposed the following question: “Fix \(n\) and suppose that the integral of \(P\) against the volume form is independent of the metric for all compact smooth manifolds of dimension \(n\). Then does there exist a universal constant \(c\) which is independent of \(M\) so that this integral is \(c\chi(M)\) where \(\chi\) is the Euler characteristic?”. The answer to this question is indeed yes and there is a local version available – there is a \(1\)-form valued invariant \(Q\) so that \(P=\delta Q+c\cdot\text{Pfaff}\) where \(\text{Pfaff}\) is the Pfaffian which expresses the integrand of the Chern-Gauss-Bonnet formula in terms of curvature and where \(Q\) is a suitable 1-form valued local invariant [P. B. Gilkey, Invariance theory, the heat equation and the Atiyah-Singer index theorem. 2nd ed. Boca Raton, FL: CRC Press (1995; Zbl 0856.58001)].
The present paper poses a slightly different question. One supposes that the integral of \(P\) against the volume form only depends on the conformal class of the metric for all compact smooth manifolds of dimension \(n\) and that the invariant \(P\) depends only on the curvature tensor and not on the covariant derivatives of the curvature tensor. One then can conclude that there is a constant \(c\) so that \(P-c\cdot\text{Pfaff}\) is a scalar conformal invariant; this settles in the affirmative a conjecture by Deser-Schwimmer (in this restricted category). Applications to the \(Q\)-curvature of Branson are then given.

MSC:

53C20 Global Riemannian geometry, including pinching
53A30 Conformal differential geometry (MSC2010)
57R20 Characteristic classes and numbers in differential topology

Citations:

Zbl 0856.58001
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References:

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