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

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 i...

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Published inDifferencialʹnaja geometrija mnogoobrazij figur Vol. 55; no. 55(1); pp. 34 - 44
Main Author Kuleshov, A. V.
Format Journal Article
LanguageEnglish
Russian
Published Immanuel Kant Baltic Federal University 2024
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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.
AbstractList 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.
Author Kuleshov, A. V.
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Cites_doi 10.5922/0321-4796-2023-54-2-1
10.1007/bf01084960
10.1007/s10958-006-0233-4
10.1007/978-3-662-02950-3
10.1134/s199508020801006x
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Snippet 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...
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StartPage 34
SubjectTerms canonical form
frame bundle
jet
smooth manifold
Title On the scalar components of the canonical form on higher order frame bundles
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Volume 55
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