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On curvatures of linear frame bundles with naturally lifted metrics. (English) Zbl 1141.53020

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.

MSC:

53C10 \(G\)-structures
53C20 Global Riemannian geometry, including pinching
53C24 Rigidity results
58A20 Jets in global analysis

Citations:

Zbl 0378.53016
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