With the maturity of depth sensors, point clouds have received increasing attention in various 3D safety-critical applications, while deep point cloud learning models have been shown to be vulnerable to adversarial attacks. Most existing 3D attackers rely on implicit global distance losses to perturb whole points, failing to restrict the proper 3D geometry as point clouds are highly structured. To this end, in this paper, we propose a novel Wavelet Patches Attack (WPA), which leverages local spectral attributes to identify curvature-aware patches for hiding imperceptible perturbations aligned with their local geometric characteristics. Specifically, WPA first transforms the point cloud into the spectral domain using a wavelet operator, obtaining potential geometric structures in different local regions. Each wavelet corresponds to different curvature contexts of local points. Then, by decomposing the 3D object with different curvature-aware levels through the wavelet coefficients, we can perceive the local geometric characteristics and get various curvature-consistent patches. At last, based on the curvature variations of patches, WPA introduces two-type perturbations along the tangent plane and normal vector direction to hide imperceptible noise in slow- and fast-variation patches for preserving the geometric-sensitive local characteristics of smoothness and sharpness, respectively. Experiments demonstrate the superior imperceptibility of our attack method, achieving favorable results on existing 3D classification models while exhibiting robust resistance to various defense mechanisms.