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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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