Differential geometric connection of finite order on composite manifold, infinitesimal polygonal loops and generalized Fermi-Walker transport (Q1198546)

A composite manifold \(L_ m(B_ n)\) (similar to a fiber bundle) is considered. The base space \(B_ n\) has coordinates \(y^ k\), \(k,\ell=1,\dots,n\), and the local space \(L_ m\) has coordinates \(\eta^ a\), \(a,b=1,\dots,m\); \(s\) runs over \(1,\dots,r\). A differential geometric connection of the \(r\)-th order is given by \(d\eta^ a/dt+\Gamma^ a\left(y^ k,d^ sy^ \ell/dt^ s,\eta^ b\right)=0\), where \(\Gamma^ a\) are the coefficients of the connection. (The nonlinear connection is that special case, when \(s=1\) and the \(\Gamma^ a\) are linear in \(dy^ \ell/dt\).) The holonomy group transformations belonging to the infinitesimal 4-gonal loops in \(L_ m(B_ n)\), calculated with an accuracy up to terms of the second order of smallness are already known in the literature and they are determined by the curvature vector (tensor) of the connection. Now the coordinate 4-gonal loops are replaced by polygonal loops, and the holonomy group transformations are calculated in this case, where beside the curvature vector other additional terms appear. In the second part of the paper \(L_ m(B_ n)\) is replaced by a vector bundle, and other simplifications are also investigated. Finally, as an illustration, the generalized Fermi-Walker transport is considered on a Riemannian tangent bundle.