THE HEAT KERNEL FORMULA IN A GEODESIC CHART AND SOME APPLICATIONS TO THE EIGENVALUE PROBLEM OF THE 3-SPHERE

This paper deals with a heat kernel formula in a geodesic chart with some applications to the standard n-sphere. Our emphasis will be on the special case of the 3-sphere which exhibits some identities linking spherical harmonics and certain homogeneous polynomials harmonic on IR4. In particular, we will deduce an expression for P(x)(zeta > t) where zeta is the first (random) time that the bridge process in S3 hits the south pole. Another easy consequence will be a special case of the H.P. McKean and I.M. Singer expansion of the heat kernel.