Evtushik, "Holonomy of nonlinear connections" Sib. Evtushik, "Non-linear connections of higher order" Izv. Vagner, "The theory of composite manifolds" Trudy Sem. A non-linear analogue has been found for the theorem on the holonomy group, and its statement involves not only the curvature, but also the linear hull of the distribution of horizontal cones, which replace in the non-linear case the subspace of the horizontal distribution of a linear connection. The forms of these linear connections have been found and also their structure equations. The foundations of the general theory of non-linear connections are fairly well developed and applications of some special types (see – ) have been investigated.īe the category of all bundles associated with $ X $.Ī bundle isomorphism of $ G _ ( B) $Īnd is completely characterized by this linear connection. The necessity of introducing and studying non-linear connections arose from the need to study various differential-geometric structures of higher orders (such as, for example, a Kawaguchi space). A different meaning of the term non-linear connection consists in the fact that the transfer for the fibres of a vector bundle defined by a horizontal distribution ceases to have a linear character, that is, is not a linear isomorphism of these fibres.
Invariant horizontal distribution of one kind or another. Here it is assumed that the structure in question is not identical with the classical concept of a linear connection, which is defined by a $ G $.
A differential-geometric structure defined for the category of smooth fibre spaces associated with a certain principal $ G $-īundle that determines the isomorphisms of the fibres (the parallel transfer) for the given non-linear connection along every piecewise-smooth curve in the base space of a bundle in the given category, which is compatible with the isomorphism of the corresponding fibres of the principal $ G $-īundle.
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