The Wikipedia entry on covariance estimation states that the sample covariance matrix (SCM), viewed in $\mathbb{R}^{p \times p}$ is an unbiased and efficient estimator, but w hen viewed intrinsically in the space of positive definite matrices, $\mathcal{S}^{p.d.}_n$, it is a biased and inefficient estimator.
How can the estimator be both biased and unbiased?
My understanding of bias is that repeated estimations performed with an unbiased estimator will converge to the true population value, in other words the average (or expected value) SCM of a set of SCMs calculated on a set of $p \times n$ samples from the same underlying distribution will be close to the true covariance matrix.
How does this change with a intrinsic or extrinsic view of the estimator? The only thing I can see is that the average SCM would be calculated differently in these different geometries.