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Locally symmetric space structures on the tangent
bundle

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*V. Oproiu and N. Papaghiuc*

**E-mail:** voproiu@uaic.ro, npapag@math.tuiasi.ro
**Abstract.** One studies the conditions under which the tangent
bundle $TM$ of a Riemannian manifold $(M,g)$, or a tube around the zero
section in $TM$, endowed with an appropriate metric $G$, is a locally symmetric
space. Locally, the matrix of $G$ is obtained as a linear combination of
the matrix associated to $g$ and a matrix of rank one, the coefficients
depending on the energy density only. The base manifold $(M,g)$ must be
a space form and $(TM,G)$ must have a structure of K\"ahler Einstein manifold.

**AMSclassification.** 53C15, 53C07, 53C55

**Keywords.** Tangent bundle, K\"ahler manifold, Einstein manifold,
locally symmetric structure