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Differential Geometry of Manifolds

2024 №55(1)

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Kuleshov A.V.

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

Abstract

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.

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