In this paper, we study the notion of incremental stability for continuous-time nonlinear systems forced by some external input. Thanks to the notion of "Killing vector field", we provide a set of sufficient conditions based on a metric analysis for an input-affine system to be incrementally inputto-state stable (δISS) and show that this also implies that its lifted system is transversally ISS. We conclude by providing a design achieving incremental ISS properties for the closed-loop system by means of a state-feedback control law that possesses an infinite gain margin property.