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On a property of sinectic metrics in the tangent bundle of the Euclidean space. (Russian) Zbl 0614.53023
Let \(T(E_ r)\) be the tangent bundle of the r-dimensional Euclidean space \(E_ r\) having the metric tensor \(\delta_{ij}\) with respect to the local coordinates \((x^ i)\) and let \((G_{\alpha \beta})\) \((\alpha,\beta =1,...,2r)\) given by \((G_{\alpha \beta})=\left( \begin{matrix} a_{ij}\\ \delta_{ij}\end{matrix} \begin{matrix} \delta_{ij}\\ 0\end{matrix} \right)\) be the sinectic metric on \(T(E_ r)\) [see N. V. Talantova and A. P. Sirokov, Izv. Vyssh. Uchebn. Zaved., Mat. 1975, No.6(157), 143-146 (1975; Zbl 0317.53017)]. The author proves here that if \(a_{ij}=0\) if \(i=j\) and \(a_{ij}=-\rho x^ ix^ j\) if \(i\neq j\), then the infinitesimal isometries in \(T(E_ r)\) are given by vector fields of the type: \[ X=\xi^ i \partial /\partial x^ i+(X^{n+s}\partial s\xi^ i+\eta^ i)\partial /\partial x^{n+i}, \] where \((\xi^ i)\) defines an infinitesimal isometry of \(E_ r\), \((\eta^ i)\) defines a vector field on \(E_ r\) by the vertical lifts of the isometries of \(E_ r\). \(T(E_ r)\) endowed with such a sinectic metric is a real model of a space with constant pure dual curvature.
Reviewer: L.Maxim-Răileanu
MSC:
53B20 Local Riemannian geometry
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