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