  We consider an inference on the eigenvalues of the covariance matrix of a
multivariate normal distribution. The family of multivariate normal
distributions with a fixed mean is seen as a Riemannian manifold with Fisher
information metric. Two submanifolds naturally arises; one is the submanifold
given by fixed eigenvectors of the covariance matrix, the other is the one
given by fixed eigenvalues. We analyze the geometrical structures of these
manifolds such as metric, embedding curvature under $e$-connection or
$m$-connection. Based on these results, we study 1) the bias of the sample
eigenvalues, 2)the information loss caused by neglecting the sample
eigenvectors, 3)new estimators that are naturally derived from the geometrical
view.
