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Functional moduli of jets of Riemannian metrics. (English. Russian original) Zbl 0920.58003
Funct. Anal. Appl. 31, No. 2, 119-125 (1997); translation from Funkts. Anal. Prilozh. 31, No. 2, 58-66 (1997).
Consider the jet (i.e., the Taylor series) of a Riemannian metric defined in a neighborhood of 0 in \(\mathbb{R}^n\). Formal diffeomorphisms preserving the origin act in an obvious way on the set of such jets. The main result of the present paper is a normal form for these jets up to the action of the formal diffeomorphisms. For example, in two dimensions the normal form is \[ ds^2 = dx^2 + xy\varphi(x,y)dx dy + dy^2, \] where \(\varphi\) is a formal power series in \(x\) and \(y\).
Let \(a_k\) denote the dimension of the orbit space of \(k\)-jets of Riemannian metrics modulo formal diffeomorphisms. As an application of the above theorem, the author can show that the Poincaré series \[ p(t) = a_0 + \sum_{k=1}^\infty (a_k-a_{k-1})t^k \] is a rational function.
Reviewer: C.Bär (Freiburg)

58A20 Jets in global analysis
58D27 Moduli problems for differential geometric structures
53B20 Local Riemannian geometry
Full Text: DOI
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