Extension of Bochner-Lichnérowicz formula on spheres
Let \(\Delta\) be the usual scalar Laplacian on the standard sphere \(S^m\) in \(\mathbb R^{m+1}\). The authors construct a sequence of bilinear forms \(\text{R}_k\) which are positive and intrinsically associated with \(\Delta\) and which carry all the information on the spectrum. Let \(\lambda_i\) be the eigenvalues of \(\Delta\). If \(f\) is an eigenfunction associated to the eigenvalue \(\lambda\), and if \(g\) is an arbitrary smooth function, then \[ R_k(f,g)=\lambda(\lambda-\lambda_1)...(\lambda-\lambda_k)\int_{S^m}{fg\, dvol}\,. \] The authors extend to the full spectrum the usual Bochner-Lichnerowicz formula giving a lower bound for the non-zero eigenvalue; this bound is sharp on the sphere.
scientific article; zbMATH DE number 5012712