Bronshtejn, I. U.; Chernij, V. F. A criterion for the existence of a continuous invariant section of an affine extension. (Russian) Zbl 0706.55012 Ukr. Mat. Zh. 42, No. 8, 1146-1151 (1990). The authors’ main result can be stated as follows: Let (X,p,B) be a vector bundle of finite rank with compact base space B. Suppose that p: (X,\({\mathbb{R}},\pi)\to (B,{\mathbb{R}},\rho)\) is a linear extension which has no nontrivial bounded motions. Let f: \(B\to X\) be a continuous section of (X,p,B). Then an affine extension p: (X,\({\mathbb{R}},\pi_ f)\to (B,{\mathbb{R}},\rho)\) has a continuous invariant section iff \[ \int^{\infty}_{-\infty}\sigma_*(\rho^ t(b))(f(\rho^ t(b))dt=0 \] (b\(\in B\); \(\pi \circ \rho^ t=\rho^ t\circ p)\) for each continuous section \(\sigma_*: B\to X^*\) which is invariant w.r.t. the linear extension \(p^*: (X^*,{\mathbb{R}},\pi^*)\to (B,{\mathbb{R}},\rho)\). Reviewer: J.Szilasi Cited in 1 Review MSC: 55R99 Fiber spaces and bundles in algebraic topology 37-XX Dynamical systems and ergodic theory Keywords:vector bundle; linear extension; invariant section PDFBibTeX XMLCite \textit{I. U. Bronshtejn} and \textit{V. F. Chernij}, Ukr. Mat. Zh. 42, No. 8, 1146--1151 (1990; Zbl 0706.55012)