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A harmonic morphism between Riemannian manifolds and is by definition a continuous mappings which pulls back harmonic functions. It is assumed that dim dim, since otherwise every harmonic morphism is constant. It is shown that a harmonic morphism is the same as a harmonic mapping in the sense of Eells and Sampson with the further property of being semiconformal, that is, a conformal submersion of the points where vanishes. Every non-constant harmonic morphism is shown to be an open mapping....
We study local reflections with respect to a curve in a Riemannian manifold and prove that is a geodesic if is a harmonic map. Moreover, we prove that the Riemannian manifold has constant curvature if and only if is harmonic for all geodesies .
Let and be two smooth vector fields on a two-dimensional manifold . If and are everywhere linearly independent, then they define a Riemannian metric on (the metric for which they are orthonormal) and they give to the structure of metric space. If and become linearly dependent somewhere on , then the corresponding Riemannian metric has singularities, but under generic conditions the metric structure is still well defined. Metric structures that can be defined locally in this way...
O. Kowalski and J. Szenthe [KS] proved that every homogeneous Riemannian manifold admits at least one homogeneous geodesic, i.eȯne geodesic which is an orbit of a one-parameter group of isometries. In [KNV] the related two problems were studied and a negative answer was given to both ones: (1) Let be a homogeneous Riemannian manifold where is the largest connected group of isometries and . Does always admit more than one homogeneous geodesic? (2) Suppose that admits linearly independent...
We characterize the boundary at infinity of a complex hyperbolic space as a compact Ptolemy space that satisfies four incidence axioms.
En 1999, M. Ledoux a démontré qu’une variété complète à courbure de Ricci positive ou nulle vérifiant une inégalité de Sobolev euclidienne était euclidienne. On présente un raccourci de la preuve. De plus nos arguments permettent un raffinement d’un résultat de B-L. Chen et X-P. Zhu à propos des variétés localement conformément plate à courbure de Ricci positive ou nulle. Enfin, on étudie ce qui se passe lorsque l’hypothèse sur la courbure de Ricci est remplacée par une hypothèse sur la courbure...
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