zbMATH — the first resource for mathematics

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).

53C05 Connections (general theory)
53C10 \(G\)-structures
58A20 Jets in global analysis
53A20 Projective differential geometry
58H05 Pseudogroups and differentiable groupoids
Full Text: DOI
[1] Bleecker, D., Gauge theory and variational principles, (1981), Addison Wesley Reading · Zbl 0481.58002
[2] Eck, D.J., Gauge natural bundles and generalized gauge theories, Memoirs of the amer. math. soc., 33, n. 247, (1981) · Zbl 0493.53052
[3] Garcia Perez, P., Gauge algebras, curvature and symplectic structure, J. diff. geo., 12, 209-227, (1977) · Zbl 0404.53033
[4] Garcia Perez, P.; Munoz Masque, J., Differential invariants on the bundles of G-structures, (), 177-201, n. 1410 · Zbl 0824.53026
[5] Kibble, T.W., Lorentz invariance and gravitational field, J. math. phys., 2, 212-221, (1961) · Zbl 0095.22903
[6] Kobayashi, S.; Nomizu, K., ()
[7] Koszul, J.L., Lectures on fibre bundles and differential geometry, (1960), Institute of fundamental Research Bombay · Zbl 0244.53026
[8] Kumpera, A.; Kumpera, A., Invariants différentiels d’un pseudogroupe de Lie II, J. diff. geo., J. diff. geo., 10, 347-416, (1975) · Zbl 0346.58012
[9] Munoz Masque, J., Formes de structure et transformations infinitésimales de contact d’order superieur, C.R. acad. sc., 298, 185-188, (1984), Série I, n. 8 · Zbl 0598.58002
[10] Sternberg, S., Lectures on differential geometry, (1964), Prentice-Hall Englewood Cliffs, N.J · Zbl 0129.13102
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.