We present a novel formulation of video stabilization in the space of geometric transformations. With the setting of the Riemannian metric, the optimized smooth path is cast as the geodesics on the Lie group embedded in transformation space. While solving the geodesics has a closed-form expression in a certain space, path smoothing can be easily implemented by using geometric interpolation, rather than optimizing any space-time energy function. Specially, by using the geodesic solution in the space of rigid transformations, our approach even gains speedup 10× faster than state-of-the-art methods for path smoothing and motion compensation, and guarantees no extra distortion drawn into the stabilized frames. The experiments demonstrate the efficiency and effectiveness of our algorithm on stabilizing a variety of shaky videos.