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Differential invariants on the bundles of linear frames. (English) Zbl 0764.53017
Let $$X$$ be an $$n$$-dimensional connected $$C^ \infty$$-manifold, $$L(X)$$ the bundle of the linear frames of $$X$$ and $$J^ r(L(X))$$ its $$r$$-jet prolongation with the local coordinates $$(x_ i,z^{ij}_ \alpha)$$, $$|\alpha|=0,1,2,\dots,r$$. Given a vector field $$D(u_ i)$$ on $$X$$, its natural lift is defined on $$L(X)$$: $$\widetilde D(u_ i,u_{ij})$$ with $$u_{ij}=\sum_ h z_{h_ j}\partial u_ i/\partial x_ h$$, and the $$r$$-order infinitesimal contact transformation $$\widetilde D_{(r)}$$ ($$r$$-order prolongation) on $$J^ r(L(X))$$. A differentiable function $$f: J^ r(L(X))\to\mathbb{R}$$ is said to be an $$r$$-order differential invariant if $$\widetilde D^{(r)}f=0$$ for every vector field $$D$$ on $$X$$, and $$r$$-order invariant under diffeomorphisms if $$f\circ\tilde\tau^{(r)}=f$$ for every diffeomorphism $$\tau$$ of $$X$$, where $$\tilde\tau^{(r)}$$ is the $$r$$-jet prolongation of the natural lift $$\tilde\tau$$ of $$\tau$$ to $$L(X)$$. The set of $$r$$-order invariants under diffeomorphisms constitutes a subring of the ring of $$r$$-order differential invariants, $$A_ r'\subset A_ r$$ (Proposition 5.2). The span $${\mathcal M}^ r$$ of all vector fields $$\widetilde D_{(r)}$$ is an involutive $$n\binom{n+r+1}{n+r}$$-dimensional distribution of $$J^ r(L(X))$$ whose first integrals generate the ring $$A_ r$$ (Theorem 2.2).
For the vector fields $$\mathbb{D}_ j=\sum_ i z_{ij}\hat\partial/\partial x_ i$$, where $$\hat\partial/\partial x_ i$$ are the operators of total differentiation, the coefficients $$f^ i_{jk}$$ from the formula $[\mathbb{D}_ j\mathbb{D}_ k]+\sum_ i f^ i_{jk}\mathbb{D}_ k=0$ are first order differential invariants (Proposition 4.5) and the derivatives $f^ i_{j_ 1\cdots j_ m k\ell}=\mathbb{D}_{j_1}\cdots\mathbb{D}_{j_ m}(f^ i_{k\ell}),\quad 0\leq m\leq r-1, \ j_ 1\geq\cdots\ge j_ m\geq k>\ell,$ constitute a local basis $$F_ r$$ of $$A_ r$$ (Theorem 4.8).
Hence it follows the condition of formal equivalence for complete parallelisms (Corollary 5.6): two frames $$s=(D^ j)$$ and $$\bar s=(\overline D^ j)$$ on $$X$$ may be transformed one in the other $$(\exists\tau,\tau\cdot D^ j=\overline D^ j)$$ if and only if the functions $$F_ r$$ coincide on the sections $$j^ r s$$ and $$j^ r\bar s$$ for all $$r\in\mathbb{N}$$ [see also Theorem 4.1 in S. Sternberg, Lectures on differential geometry. Englewood Cliffs: Prentice-Hall (1964; Zbl 0129.13102)]. The global strutures of the rings $$A_ r$$ and $$A_ r'$$ for (1) $$X$$ non-orientable, (2) $$X$$ orientable but non-reversible and (3) $$X$$ reversible are described (Theorem 5.7).

MSC:
 53C05 Connections (general theory) 53C10 $$G$$-structures 58A20 Jets in global analysis 53A20 Projective differential geometry 58H05 Pseudogroups and differentiable groupoids
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References:
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