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Let M be a complete Riemannian manifold and N a complete noncompact Riemannian manifold. Let ϕ:M→N be a surjective harmonic morphism. We prove that if N admits a subharmonic function with finite Dirichlet integral which is not harmonic, and ϕ has finite energy, then ϕ is a constant map. Similarly, if f is a subharmonic function on N which is not harmonic and such that |df| is bounded, and if ∫M|dϕ|

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.

Let (Mn,g) be a closed Riemannian manifold and N a warped product manifold of two space forms. We investigate geometric properties by the spectra of the Jacobi operator of a harmonic map ϕ:M→N. In particular, we show if N is a warped product manifold of Euclidean space with a space form and ϕ,ψ:M→N are two projectively harmonic maps, then the energy of ϕ and ψ are equal up to constant if ϕ and ψ are isospectral. Besides, we recover and improve some results by Kang, Ki, and Pak (1997) and Urakawa (1989).

In this paper, we first consider the disjoint union of two fuzzy metric spaces and non-existence of fuzzy contractive maps. We prove that there exists a natural fuzzy metric on the disjoint union of two fuzzy metric spaces. We also show that there is a fuzzy metric space which does not admit any fuzzy contractive map. Second, we introduce the notion of a fuzzy diameter for fuzzy metric spaces and obtain some properties on it and its relations to classical diameter for compact metric spaces. Moreover, using the notion of fuzzy diameter, we construct a fuzzy metric on the disjoint union of two fuzzy metric spaces. Finally, we consider the fuzzy product metric space of two fuzzy metric spaces and prove that there are fuzzy metrics on the product space such that the inclusions into the product space become fuzzy isometric embeddings. This shows that there is a family of fuzzy isometric embeddings from a fuzzy metric space into another fuzzy metric space. [ABSTRACT FROM AUTHOR]

In this article we prove a Liouville type theorem for p-harmonic morphisms. We show that if φ: M→N is a p-harmonic morphism (p≥2) from a complete noncompact Riemannian manifold M of nonnegative Ricci curvature into a Riemannian manifold N of nonpositive scalar curvature such that the p-energy Ep(φ), or (2p−2)-energy E2p−2(φ) is finite, then φ is constant. [ABSTRACT FROM AUTHOR]

We show that given $n$ and $D, v >0$, there exists a positive number $ \epsilon = \epsilon (n,D,v)> 0$ such that if a closed $n$-manifold $M$ satisfies $Ric(M) \ge -\epsilon , diam(M) \le D$ and $vol(M) \ge v$, then $\pi _{1}(M)$ is almost abelian. [ABSTRACT FROM AUTHOR]