We study for a class of symmetric Lévy processes with state space ℝ ⁿ the transition density p _t (x) in terms of two one-parameter families of metrics, (d _t) _t>0 and (δ _t) _t>0. The first family of metrics describes the diagonal term p _t (0); it is induced by the characteristic exponent ψ of the Lévy process by d _t(x,y)=√tψ(x-y). The second and new family of metrics δ _t relates to √tψ through the formula,where F denotes the Fourier transform. Thus we obtain the following "Gaussian" representation of the transition density:corresponds to a volume term related to p _t(0) and where an √tψ. This gives a complete and new geometric, intrinsic interpretation of p _t(x).