  In this paper the issue of filtering and smoothing in continuous discrete
time is studied when the state variable evolves in some submanifold of
Euclidean space, which may not have the usual Lebesgue measure. Formal
expressions for prediction and smoothing problems are derived, which agree with
the classical results except that the formal adjoint of the generator is
different in general. For approximate filtering and smoothing the projection
approach is taken, where it turns out that the prediction and smoothing
equations are the same as in the case when the state variable evolves in
Euclidean space. The approach is used to develop projection filters and
smoothers based on the von Mises-Fisher distribution.
