On moduli spaces for finite-order jets of linear connections

Abstract

We describe the ringed-space structure of moduli spaces of jets of linear connections (at a point) as orbit spaces of certain linear representations of the general linear group.Then, we use this fact to prove that the only (scalar) differential invariants associated to linear connections are constant functions, as well as to recover various expressions appearing in the literature regarding the Poincaré series of these moduli spaces.

Poincaré Function for Moduli of Differential-Geometric Structures

“…The bundle E is however algebraic and the conclusion of Theorem 1 holds true. Indeed, scalar differential invariants of 0-th order are rational invariants of the general linear group on the space of torsion tensors (note that scalar polynomial differential invariants are only constants [11]). For n ≥ 3 these also generate invariant derivations, whence a Lie-Tresse generation property (first order invariants should be used for n = 2 to get this).…”

A remark on the invariant theory of real Lie groups

“…Nevertheless, this group also has the structure of a real Lie group. Recently, both the first and second main theorems for this group O p,q have been widely used in the realm of natural operations in pseudo-Riemannian geometry ( [1], [2], [6], [7]) or in the construction of moduli spaces of jets ( [4], [5]). In these settings, the point of view is that of Differential Geometry; therefore, O p,q is understood as a real Lie group and, in principle, it is not clear why the invariants in the "differentiable" sense of this non-compact and non-connected Lie group should coincide with those invariants computed in the "algebraic" sense, where the action of the affine R-group encodes information, not only of real points, but also of complex ones.…”

On Invariant Operations on a Manifold with a Linear Connection and an Orientation

“…As for the existence of smooth sections, let us choose a local coordinate system centered at p. For any given formal development j ∞ p ∇, the proof of [20] (Thm. 3.6) shows how these coordinates define a global section σ m of φ m that passes through j m p ∇.…”