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This paper presents a novel approach that achieves complete matching of 3D dynamic surfaces. Surfaces are captured from multi-view video data and represented by sequences of 3D manifold meshes in motion (3D videos). We propose to perform dense surface matching between 3D video frames using geodesic diffeomorphisms. Our algorithm uses a coarse-to-fine strategy to derive a robust correspondence map, then a probabilistic formulation is coupled with a voting scheme in order to obtain local unicity of matching candidates and a smooth mapping. The significant advantage of the proposed technique compared to existing approaches is that it does not rely on a color-based feature extraction process. Hence, our method does not lose accuracy in poorly textured regions and is not bounded to be used on video sequences of a unique subject. Therefore our complete surface mapping can be applied to: (1) texture transfer between surface models extracted from different sequences, (2) dense motion flow estimation in 3D video, and (3) motion transfer from a 3D video to an unanimated 3D model. Experiments are performed on challenging publicly available real-world datasets and show compelling results.