Harmonicity of horizontally conformal maps and spectrum of the Laplacian

We discuss the harmonicity of horizontally conformal maps and their relations with the spectrum of the Laplacian. We prove that if φ : M → N is a horizontally conformal map such that the tension field is divergence free, then φ is harmonic. Furthermore, if N is noncompact, then φ must be constant. Also we show that the projection of a warped product manifold onto the first component is harmonic if and only if the warping function is constant. Finally, we describe a characterization for a horizontally conformal map with a constant dilation preserving an eigenfunction. We describe here some characterizations for the harmonicity of horizontally con- formal maps. In particular, we consider the projections of a warped product man- ifold onto each component manifold. Those are examples of horizontally confor- mal maps. We show that the projection of a warped product manifold onto the first component is a horizontally conformal map if and only if the warping function is constant. Finally, we consider the spectrum of the Laplacian and its relations with horizontally conformal maps. In (6), Gilkey and Park studied the spectrum of the Laplacian and Riemannian submersions. They proved that a Riemannian submersion φ : M → N commutes with the Laplacian if and only if φ ∗ preserves the eigenfunctions of the Laplacian. In (10), the author showed that, for horizontally conformal maps, a similar result hold. If a horizontally conformal map preserves an eigenfunction, then the dilation of the horizontally conformal map is given by the square root of the ratio of eigenvalues or a geometric identity must hold. Throughout, every manifold is connected and smooth, and a compact manifold is assumed to be compact without boundary otherwise stated.