Authors’ abstract: We study for a class of symmetric Lévy processes with state space \(\mathbb{R}^n\) the transition density \(p_ {t} (x)\) in terms of two one-parameter families of metrics, \((d_ {t})_ { t>0}\) and \((\delta _ {t} )_ { t>0}\). The first family of metrics describes the diagonal term \(p_ {t} (0)\); it is induced by the characteristic exponent \(\psi \) of the Lévy process by \(d_ {t} (x,y)=\sqrt{t \psi (x-y)}\). The second and new family of metrics \(\delta _ {t}\) relates to \(\sqrt{t \psi }\) through the formula \[ \exp (-\delta _ {t}^ {2} (x,y))={\mathcal F} \left [\frac{e^ { -t \psi }}{ p_ { t} (0)}\right ] (x-y), \] where \(\mathcal F\) denotes the Fourier transform. Thus we obtain the following “Gaussian” representation of the transition density: \(p_ { t} (x)=p_ { t} (0)e^ { -\delta _ { t}^ { 2} (x,0)}\) corresponds to a volume term related to \(\sqrt{t \psi }\) and where an “exponential” decay is governed by \(\delta _ { t}^ { 2}\). This gives a complete and new geometric, intrinsic interpretation of \(p_ { t} (x)\).