The ability to remember and recognize other individuals is crucial for complex social behaviors. Although the hippocampus is known to be fundamental for social memory (Hitti & Siegelbaum, 2014), how this region represents and supports the two processes of social recognition - familiarity and recollection - remains unknown. We used microendoscopic calcium imaging to measure the activity of hippocampal dorsal CA2 pyramidal neurons (dCA2) - a region known for its importance in social memory - in mice interacting with novel and familiar conspecifics in different spatial contexts. Through a decoding analysis, we reconstructed the representational geometry of social and spatial variables in a series of social interaction experiments. We found that both social (familiarity and individual identity of the encountered mouse) and spatial (position of the encounter) variables are decodable from CA2 activity, and that the decoding performance for familiarity positively correlates with behavioral preference for novel animals. Importantly, CA2 represents familiarity as an abstract variable, suggesting that representations of novel and familiar individuals lie on distinct but parallel manifolds in the neural activity space. By comparing the representations of familiar and novel animals, we found that familiarity distorts the representational geometry by increasing its dimensionality, enhancing social-spatial discrimination at the expense of generalization. Finally, we confirmed our geometrical interpretation by showing that a statistical model based on geometrical reasoning is able to reproduce the experimental results. Thus, dCA2 population activity supports social recognition memory through an abstract representation of familiarity and a change in the geometrical properties of recollected social identities.