Functional moduli of jets of Riemannian metrics. (English. Russian original) Zbl 0920.58003

Shmelev, A. S.

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)

MSC:

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

Keywords:

normal form; Poincaré series; jet; Riemannian metric; moduli space

References:

[1]	V. I. Arnold, Mathematical Problems in Classical Physics, Appl. Math. Sci., Vol. 100, Springer-Verlag, Berlin-New York, 1992.
[2]	A. Tresse, ”Sur les invariants differentiels des groupes continus des transformations,” Acta Math.,18 (1894). · JFM 26.0340.01
[3]	A. S. Shmelev, ”Differential invariants of some differential-geometric structures,” Trudy Mat. Inst. Steklov,209, 234–267 (1995). · Zbl 0886.58012
[4]	Sh. Kobayashi and K. Nomizu, Foundations of Differential Geometry [Russian translation], Vol. 1, Nauka, Moscow, 1981. · Zbl 0526.53001