Kowalski, O.; Sekizawa, M. On curvatures of linear frame bundles with naturally lifted metrics. (English) Zbl 1141.53020 Rend. Semin. Mat., Torino 63, No. 3, 283-295 (2005). The authors study some special natural metrics, generalizations of the diagonal metric by K.-P. Mok [J. Reine Angew. Math. 302, 16–31 (1978; Zbl 0378.53016)], on the linear frame bundle \(LM\) over a Riemannian manifold \((M,g).\) They derive explicit formulae for the Levi-Civita connection and the curvature tensor by lengthy direct calculations. All types of curvatures on \(LM\) are determined in the particular case if \((M,g)\) is of a constant sectional curvature. In more detail. Let \(X^h, X^{v,\mu}\) \((\mu=1,\ldots,n)\) be the horizontal and the vertical lifts on \(LM\) of a vector field \(X\) on \(M.\) Then the natural metrics \(G\) on \(LM\) under consideration are \[ G(X^h,Y^h)=\sum w_{\alpha\beta}\omega^{\alpha}(X)\omega^{\beta}(Y), \]\[ G(X^h,Y^{v,\mu})=\sum c^{\mu}w_{\alpha\beta}\omega^{\alpha}(X)\omega^{\beta}(Y), \]\[ G(X^{v,\lambda},Y^{v,\mu})=\sum c^{\lambda\mu}w_{\alpha\beta}\omega^{\alpha}(X)\omega^{\beta}(Y), \] where \(\omega^1,\ldots,\omega^n\) is the dual frame to a linear frame \(u_1,\ldots,u_n\) on \(M,\) \(w_{\rho\sigma}=g(u_{\rho},u_{\sigma})\) and \(c^{\mu}, c^{\lambda\mu}\) are constants. Reviewer: Jan Chrastina (Brno) Cited in 7 Documents MSC: 53C10 \(G\)-structures 53C20 Global Riemannian geometry, including pinching 53C24 Rigidity results 58A20 Jets in global analysis Citations:Zbl 0378.53016 PDFBibTeX XMLCite \textit{O. Kowalski} and \textit{M. Sekizawa}, Rend. Semin. Mat., Torino 63, No. 3, 283--295 (2005; Zbl 1141.53020) Full Text: EuDML