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Invariant theory and calculus for conformal geometries. (English) Zbl 1004.53010
A conformal geometry on a manifold \(M\) is an equivalence class of Riemannian metrics on \(M\) that differ only by multiplication with a scalar function. If \(\dim M\geq 3\), then conformal geometries possess local invariants. Here, a local invariant is an invariant polynomial \(I(g)\) in the coefficients of a metric \(g\) and their derivatives such that \(I(\Omega^2g)=\Omega^u I(g)\). If \(d\) is the degree of the polynomial \(I\) and \(k\) is the total number of derivatives involved, then \(u=-(2d+k)\).
By recent work of Fefferman and Graham, for odd-dimensional conformal manifolds, all such invariants can be obtained using Fefferman’s ambient metric construction as described by T. N. Bailey, M. G. Eastwood and C. R. Graham in [Ann. Math. (2) 139, No. 3, 491-552 (1994; Zbl 0814.53017)]. In even-dimensional conformal geometry, the ambient metric construction is obstructed, and the methods above only give a finite number of independent invariants.
The paper under review uses T. Y. Thomas’ tractor calculus, in a way similar to the author’s approach in [Math. Ann. 306, 513-538 (1996; Zbl 0904.53014)]. The author presents a method for the construction of so-called quasi-Weyl invariants and proves that in each fixed dimension, all local invariants are quasi-Weyl except for a finite number of values of \(d\) and \(k\). In odd dimensions, the quasi-Weyl invariants above are in fact Weyl invariants, which are easier to describe.

53A30 Conformal differential geometry (MSC2010)
53A35 Non-Euclidean differential geometry
Full Text: DOI
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