Harmonicity of horizontally conformal maps and spectrum of the Laplacian
Let \(\phi:(M,g_M)\to(N,g_N)\) be a smooth map between two Riemannian manifolds. It is harmonic if the tension \(\tau(\phi)=\text{tr}(\nabla d\phi)\in\Gamma(\phi^*TN)\) vanishes. The map \(\phi\) is said to be horizontally conformal if \(\phi_*\) is conformal on the orthogonal complement to \(\text{Ker}(\phi_*)\subset TM\). It is a harmonic morphism if \(\phi\) is simultaneously harmonic and horizontally conformal. The following statements are proved: 1. If \(M\) is compact, \(\text{div}(\tau(\phi))=0\) and \(\phi\) is horizontally conformal, then \(\phi\) is a harmonic morphism. If moreover \(N\) is noncompact, then \(\phi\) is constant. 2. If \(M=N\times_{\alpha}F\) is a warped product, \(g_M=g_N+\alpha^2g_F\), with \(\phi\) being the natural projection, then \(\phi\) is harmonic iff \(\alpha=\text{const}\). 3. Let \(M\) be compact and \(\phi\) horizontally conformal with a constant dilation \(\rho\). Suppose there exists a non-constant eigenfunction \(f\) of the Laplacian on \(N\) with eigenvalue \(\lambda\) that is mapped to an eigenfunction \(\phi^*f\) on \(M\) with eigenvalue \(\mu\). If \(\rho^2\neq\mu/\lambda\), then \(\int_M(\phi^*f)\text{div}(\kappa)dv_M=0\), where \(\kappa\) is the mean curvature of the \(\phi\)-fibers.
scientific article; zbMATH DE number 1779154