  Covariance representations are developed for the uniform distributions on the
Euclidean spheres in terms of spherical gradients and Hessians. They are
applied to derive a number of Sobolev type inequalities and to recover and
refine the concentration of measure phenomenon, including second order
concentration inequalities. A detail account is also given in the case of the
circle, with a short overview of H\"offding's kernels and covariance identities
in the class of periodic functions.
