The Laplacian and the heat kernel acting on differential forms on spheres
Consider the identity \(S^n\equiv SO(n+1)/SO(n)\) defined by the canonical action of \(SO(n+1)\) on the \(n\)-sphere. Let \(T_eSO(n)\equiv\mathfrak{g}=\mathfrak{so}(n+1)\) be endowed with the inner product \((\cdot,\cdot)=-B(\cdot,\cdot)\), where \(B\) is the Killing form. Let \(\mathfrak{k}=\mathfrak{so}(n)\subset\mathfrak{g}\), and let \(\mathfrak{m}\) denote its orthogonal complement. Thus \(T_{[e]}S^n\equiv\mathfrak{m}\). Let \(E_{i,j}\) be the \((n+1)\times(n+1)\)-matrix of which the \((i,j)\)-entry is \(1\) and the other entries are \(0\). Then \(e_{i,j}=(2(n-1))^{-1/2}(E_{i,j}-E_{j,i})\) (\(1\leq i<j\leq n\)) is a canonical orthonormal frame of \(\mathfrak{g}\). This frame is also denoted by \((e_1,\dots,e_{n(n+1)/2})\), or by \((e_{\mathfrak{m}1},\dots,e_{\mathfrak{m}n},e_{\mathfrak{k}1}\dots,e_{\mathfrak{k}n(n-1)/2})\) according to the decomposition \(\mathfrak{g}=\mathfrak{m}\oplus\mathfrak{k}\). The corresponding dual frame of \(\mathfrak{g}^*\) is denoted by \((e^1,\dots,e^{n(n+1)/2})\), or \((e^{\mathfrak{m}1},\dots,e^{\mathfrak{m}n},e^{\mathfrak{k}1}\dots,e^{\mathfrak{k}n(n-1)/2})\). The superindex or subindex \(L\) is added to denote the corresponding left invariant vector fields or \(1\)-forms on \(SO(n+1)\). Then the Laplacian of \(SO(n+1)\) acting on functions is given by \(\Delta_0^{SO(n+1)}=-\sum_ie_i^Le_i^L\). The paper studies the lift of the Laplacian \(\Delta^{S^n}\), acting on differential forms, by the Riemannian submersion \(\pi:SO(n+1)\to S^n\). Let \(\tilde\omega=\sum_\alpha e^{\mathfrak{m}\alpha}_L\cdot\Omega_\alpha\) be the lift of a differential form \(\omega\), where the usual multi-index notation is used with \(\alpha=(\alpha_1,\dots,\alpha_{|\alpha|})\), and the coefficients \(\Omega_\alpha\) are functions on \(SO(n+1)\). The first main result of the paper states that \(\widetilde{\Delta^{S^n}\omega}=\sum_\alpha e^{\mathfrak{m}\alpha}_L\cdot\Delta_0^{SO(n+1)}\Omega_\alpha\). This is satisfactory because the study of the heat operator \(e^{-t\Delta^{S^n}}\), acting on differential forms, can be reduced to the study of the heat operator \(e^{-t\Delta_0^{SO(n+1)}}\), acting on functions. In particular, by applying the Poisson summation formula for the heat kernel of \(\Delta_0^{SO(n+1)}\), proved by \textit{H. Urakawa} [Osaka J. Math. 12, 285--297 (1975; Zbl 0335.22009)], and using the Weyl integration formula, the author obtains a formula for the heat kernel of \(\Delta^{S^n}\) on the diagonal of \(S^n\times S^n\).
scientific article; zbMATH DE number 5687944
scientific article