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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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