Harmonic spaces and density functions
Let \(\mathcal{M=}\left( M,g\right) \) be an \(m\)-dimensional Riemannian manifold, \(P\in M\) and \(\exp_{P}:M_{p}\rightarrow M\) be the exponential map at \(P\). By fixing an orthonormal basis in \(M_{p}\), a local normal coordinates \(x^{1},x^{2},\dots,x^{m}\) centered at \(P\) via the exponential map is introduced. If \(\xi\) is a vector in \(M_{p}\) with coordinates \(\xi^{1},\xi^{2},...,\xi^{m} \), then the volume element in the local normal coordinates \(x^{1} ,x^{2},..,x^{m}\) is written as \(\widetilde{\Theta}_{\mathcal{M}}\left( P;\xi\right) d\xi\), where \(\widetilde{\Theta}_{\mathcal{M}}\left( P;\xi\right) \) is the density function. In polar coordinates, the density function is given by \(\Theta_{\mathcal{M}}\left( P;\xi\right) =r^{m-1}\) \(\widetilde{\Theta}_{\mathcal{M}}\left( P;\xi\right) \). A Riemannian manifold \(\mathcal{M}\) is called a harmonic space, if \(\Theta_{\mathcal{M} }\left( P;\xi\right) \) is a radial function \(\Theta_{\mathcal{M}}\left( P;r\right) \). The definition of harmonic spaces can be given in several equivalent ways (Theorem 1.3). Recall that the Jacobi operator \(J_{\mathcal{M} }\left( \xi\right) \eta\) is \(R_{\mathcal{M}}\left( \eta,\xi\right) \xi\). The authors consider harmonic spaces with the following density functions \[ \Theta_{\varepsilon}\left( m,k\right) \left( r\right) =\left\{ \begin{array} [c]{c} \sin\left( r\right) ^{m-1}\cos\left( r\right) ^{k}\text{ if } \varepsilon=+,\\ \sinh\left( r\right) ^{m-1}\cosh\left( r\right) ^{k}\text{ if } \varepsilon=-. \end{array} \right. \] In their main result (Theorem 1.8), the authors obtain estimates for averages of the eigenvalues of the (reduced) Jacobi operator and prove that when an equality holds in one of their estimates, then \(\mathcal{M}\) is locally isometric to a rank one symmetric space. The authors also prove estimates for individual eigenvalues (Theorem 1.9). Let \(\widetilde{\Theta}_{\mathcal{M}}\left( P;\xi\right) \sim1+\Sigma _{k=2}^{\infty}\mathcal{H}_{k}\left( P;\xi\right) \) be the formal Taylor series, where \(\mathcal{H}_{k}\left( P;\xi\right) \) is homogeneous of degree \(k\) in \(\xi\). They derive formulas for \(\mathcal{H}_{k}\left( P;\xi\right) ,k=1,2,...,8\) (these formulas are new for \(k=7,8\)) in terms of the operators \(J_{k}\), where \(J_{k}\left( \xi\right) =\nabla_{\xi} ^{k}J_{\mathcal{M}}\left( \xi\right) \) (Theorem 1.1). Then they simplify these formulas for harmonic spaces (Theorem 1.4) and use their Theorem 1.4 via Lemma 1.7 to reduce their estimates to \(0< k< m-1\).
scientific article; zbMATH DE number 7220543
scientific article