Increasing works for antibody design are emerging to generate sequences and structures in Complementarity Determining Regions (CDRs), but problems still exist. We focus on two of them: (i) authenticity of the generated structure and (ii) rationality of the affinity maturation, and propose GeoAB as a solution. In specific, GeoAB-Designergenerates CDR structures with realistic internal geometries, composed of a generative geometry initializer (Geo-Initializer) and a position refiner (Geo-Refiner); GeoAB-Optimizer achieves affinity maturation by accurately predicting both the mutation effects and structures of mutant antibodies with the same network architecture as Geo-Refiner. Experiments show that GeoAB achieves state-of-the-art performance in CDR co-design and mutation effect predictions, and fulfills the discussed tasks effectively.