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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.

##### MSC:
 53A30 Conformal differential geometry (MSC2010) 53A35 Non-Euclidean differential geometry
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##### References:
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