  We prove a central limit theorem (CLT) for the Frechet mean of independent
and identically distributed observations in a compact Riemannian manifold
assuming that the population Frechet mean is unique. Previous general CLT
results in this setting have assumed that the cut locus of the Frechet mean
lies outside the support of the population distribution. So far as we are
aware, the CLT in the present paper is the first which allows the cut locus to
have co-dimension one or two when it is included in the support of the
distribution. A key part of the proof is establishing an asymptotic
approximation for the parallel transport of a certain vector field. Whether or
not a non-standard term arises in the CLT depends on whether the co-dimension
of the cut locus is one or greater than one: in the former case a non-standard
term appears but not in the latter case. This is the first paper to give a
general and explicit expression for the non-standard term which arises when the
co-dimension of the cut locus is one.
