On the conformal metric of annulus in the n-dimensional Euclidean space

It is shown by the methods of symmetrization that the geodesic with respect to the conformal metric of annulus in the Euclidean space is located into a two-dimensional sector. As a consequence, the geodesic is established in the case of points located on symmetric sphere of the annulus. Exact lower bounds are proved for the conformal metric of the annulus. A distortion theorem for quasi-regular mappings is given.