We propose connectivity-preserving geometry images (CGIMs), which map a three-dimensional mesh onto a rectangular regular array of an image, such that the reconstructed mesh produces no sampling errors, but merely round-off errors. We obtain a V-matrix with respect to the original mesh, whose elements are vertices of the mesh, which intrinsically preserves the vertex-set and the connectivity of the original mesh in the sense of allowing round-off errors. We generate a CGIM array by using the Cartesian coordinates of corresponding vertices of the V-matrix. To reconstruct a mesh, we obtain a vertex-set and an edge-set by collecting all the elements with different pixels, and all different pairwise adjacent elements from the CGIM array respectively. Compared with traditional geometry images, CGIMs achieve minimum reconstruction errors with an efficient parametrization-free algorithm via elementary permutation techniques. We apply CGIMs to lossy compression of meshes, and the experimental results show that CGIMs perform well in reconstruction precision and detail preservation.