The heat kernel formula in a geodesic chart and some applications to the eigenvalue problem of the 3-sphere (Q756858)

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 \({\mathbb{R}}^ 4\). In particular, we will deduce an expression for \(P_ x(\zeta >t)\), where \(\zeta\) is the first (random) time that the bridge process in \(S^ 3\) hits the south pole. Another easy consequence will be a special case of the \textit{H. P. McKean jun.} and \textit{I. M. Singer} [J. Differ. Geom. 1, No.1, 43-69 (1967; Zbl 0198.443)] expansion of the heat kernel.