On the scalar components of the canonical form on higher order frame bundles

• Published in: Differencialʹnaja geometrija mnogoobrazij figur Vol. 55; no. 55(1); pp. 34 - 44
• Main Author: Kuleshov, A. V.
• Format: Journal Article
• Language: English; Russian
• Published: Immanuel Kant Baltic Federal University 2024
• Subjects: canonical form; frame bundle; jet; smooth manifold
• ISSN: 0321-4796; 2782-3229
• DOI: 10.5922/0321-4796-2024-55-1-4

A detailed obtaining of the expressions for the scalar components of the canonical form on higher order frame bundles over a smooth manifold has been done. The canonical form on the frame bundle of order p + 1 over an n-dimensional smooth manifold is a vector-valued differential 1-form with values in the tangent space to the p-th order frame bundle over the n-di­mensional arithmetical space at the unit of the p-th order differential group. The scalar components of the canonical form are its coefficients with respect to natural basis of the tangent space. For every frame, there exists a polynomial mapping representing the frame in a given local chart on the manifold. Therefore, for any tangent vector to the frame bundle there is a first order Taylor expansion of one-parametric family of poly­nomial mappings representing the tangent vector. We obtain the formulas of the scalar components from the equations for coefficients of the two expansions for some tangent vector.