The lifts of polyaffinor structures on the pure sections of a tensor bundle
type: Item
instance of: scholarly article
title: The lifts of polyaffinor structures on the pure sections of a tensor bundle
zbMATH Open document ID: 0894.53038
published in: Russian Mathematics
publication date: 1998-04-01T00:00:00Z
review text: A polyaffinor structure \(\Pi\) on a manifold \(M\) is a collection of affinor fields. A tensor field \(t\) is said to be pure with respect to \(\Pi\) if for each \(\phi \in \Pi\) there holds \( t(\phi x_1, \dots, x_q, \xi_1, \dots, \xi_p) = \dots = t(x_1, \dots, \phi x_q, \xi_1, \dots, \xi_p) = t(x_1, \dots, x_q, \phi^*\xi_1, \dots, \xi_p) = \dots = t(x_1, \dots, x_q, \xi_1, \dots, \phi^*\xi_p) \). If \(\Pi\) is rigid, i.e., in a neighborhood of each point in \(M\) there exists a frame with respect to which all affinors in \(\Pi\) have constant components, then the pure tensors of type \((p,q)\) constitute the pure tensor subbundle \(t^p_q(M)\) with respect to \(\Pi\). The author considers the vertical, horizontal and complete lifts of vector fields to the tensor bundle \(T^p_q(M)\), and finds their properties. Then along \(t^p_q(M)\) the author constructs lifts of affinors of \(\Pi\) to \(T^p_q(M)\) and studies them with the use of Tachibana operators. For example, he proves the following Theorem. Let \(\Pi\) be almost integrable. Then the complete lift and the horizontal lift of \(\Pi\) to the tensor bundle \(T^p_q(M)\) along \(t^p_q(M)\) coincide on a pure section.
Mathematics Subject Classification ID: 53C15
Mathematics Subject Classification ID: 53A45
zbMATH DE Number: 1135537
zbMATH Keywords: complete lift
zbMATH Keywords: vertical lift
zbMATH Keywords: horizontal lift
zbMATH Keywords: pure tensor field
zbMATH Keywords: tensor bundle
zbMATH Keywords: polyaffinor structure
author: A. Salimov
reviewed by: M. A. Malakhaltsev
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1135537