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On frames defined by horizontal spaces. (English) Zbl 0870.58005

The authors consider the second order holonomic frame bundle \(F^2M\) over an \(n\)-dimensional manifold \(M.\) The structure group \(G^2(n)\) of \(F^2M\) is the semi-direct product of \(GL_n(\mathbb{R})\) and \(S^2(n)\) – the additive group of symmetric bilinear mappings of \(\mathbb{R}^n\times \mathbb{R}^n\) into \(\mathbb{R}^n.\) Then the universal connection on the 1st jet prolongation in the sense of P. L. Garcia [Rend. Sem. Mat. Univ. Padova 47, 227-242 (1972; Zbl 0251.53024)]is considered, and expressed in terms of the canonical form on the 2nd non-holonomic frame bundle \(\hat F^2M.\) As an application, the authors prove a result of P. Libermann [Atti Convegno Internaz. Geometria Differenziale, Bologna, 1967, 65-82 (1970; Zbl 0231.53051)]and P.Ch. Yuen [Cah. Top. Geom. Differ. 12, 333-371 (1971; Zbl 0222.53033)]on integrability of a 2nd order semi-holonomic parallelism.
Reviewer: D.Krupka (Opava)

MSC:

58A20 Jets in global analysis
53B15 Other connections
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References:

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