[size=13.3333px]In this paper, we derive a general asymptotic implied volatility atthe first-order for any stochastic volatility model using the heat kernel expansionon a Riemann manifold endowed with an Abelian connection. This formula isparticularly useful for the calibration procedure. As an application, we obtain anasymptotic smile for a SABR model with a mean-reversion term, called λ-SABR,corresponding in our geometric framework to the Poincar ́e hyperbolic plane.When the λ-SABR model degenerates into the SABR-model, we show that ourasymptotic implied volatility is a better approximation than the classical Hagan-al expression [19]. Furthermore, in order to show the strength of this geometricframework, we give an exact solution of the SABR model with β = 0 or 1. In anext paper, we will show how our method can be applied in other contexts suchas the derivation of an asymptotic implied volatility for a Libor market modelwith a stochastic volatility ([23]).