This paper proposes a stealthy integrity attack generation methodology for a class of nonlinear cyber-physical systems. Geometric control theory and stability theory of incremental systems are used to design an attack generation scheme with stealthiness properties. An attack model is proposed as a closed-loop dynamical system with an arbitrary input signal. This model is developed based on a controlled invariant subspace that results from geometric control theory and is decoupled with the system outputs and the nonlinear function. The presence of the arbitrary signal in the attack model provides an additional degree of freedom and constitutes a novel component compared with existing results. The stealthiness property of the attack model is rigorously investigated based on the incremental stability of the closed-loop control system, and the incremental input-to-state stability of the anomaly detector. As a result, a sufficient condition in terms of the initial condition of the attack model is derived to guarantee stealthiness. Finally, a case study is presented to illustrate the effectiveness of the developed attack generation scheme.