We study the problem of characterizing functions, which when applied entrywise, preserve Loewner positivity on distinguished submanifolds of the cone of positive semidefinite matrices. Following the work of Schoenberg and Rudin (and several others), it is well-known that entrywise functions preserving positivity in all dimensions are necessarily absolutely monotonic. However, there are strong theoretical and practical motivations to study functions preserving positivity in a fixed dimension $n$. Such characterizations are known only in the $n=2$ case.