  Normalizing Flows (NFs) are powerful and efficient models for density
estimation. When modeling densities on manifolds, NFs can be generalized to
injective flows but the Jacobian determinant becomes computationally
prohibitive. Current approaches either consider bounds on the log-likelihood or
rely on some approximations of the Jacobian determinant. In contrast, we
propose injective flows for parametric hypersurfaces and show that for such
manifolds we can compute the Jacobian determinant exactly and efficiently, with
the same cost as NFs. Furthermore, we show that for the subclass of star-like
manifolds we can extend the proposed framework to always allow for a Cartesian
representation of the density. We showcase the relevance of modeling densities
on hypersurfaces in two settings. Firstly, we introduce a novel Objective
Bayesian approach to penalized likelihood models by interpreting level-sets of
the penalty as star-like manifolds. Secondly, we consider Bayesian mixture
models and introduce a general method for variational inference by defining the
posterior of mixture weights on the probability simplex.
