Matches in DBpedia 2014 for { ?s ?p "A (smooth) map φ:M→N between Riemannian manifolds M and N is called harmonic if it is a critical point of the Dirichlet energy functionalThis functional E will be defined precisely below—one way of understanding it is to imagine that M is made of rubber and N made of marble (their shapes given by their respective metrics), and that the map φ:M→N prescribes how one "applies" the rubber onto the marble: E(φ) then represents the total amount of elastic potential energy resulting from tension in the rubber. In these terms, φ is a harmonic map if the rubber, when "released" but still constrained to stay everywhere in contact with the marble, already finds itself in a position of equilibrium and therefore does not "snap" into a different shape.Harmonic maps are the 'least expanding' maps in orthogonal directions.Existence of harmonic maps from a complete Riemannian manifold to a complete Riemannian manifold of non-positive sectional curvature was proved by Eells & Sampson (1964)."@en }
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- Harmonic_map abstract "A (smooth) map φ:M→N between Riemannian manifolds M and N is called harmonic if it is a critical point of the Dirichlet energy functionalThis functional E will be defined precisely below—one way of understanding it is to imagine that M is made of rubber and N made of marble (their shapes given by their respective metrics), and that the map φ:M→N prescribes how one "applies" the rubber onto the marble: E(φ) then represents the total amount of elastic potential energy resulting from tension in the rubber. In these terms, φ is a harmonic map if the rubber, when "released" but still constrained to stay everywhere in contact with the marble, already finds itself in a position of equilibrium and therefore does not "snap" into a different shape.Harmonic maps are the 'least expanding' maps in orthogonal directions.Existence of harmonic maps from a complete Riemannian manifold to a complete Riemannian manifold of non-positive sectional curvature was proved by Eells & Sampson (1964).".