Harmonic Morphisms between Riemannian Manifolds

By P. Baird, J. Wood. 2003

671 citations recorded by the source corpus.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/f5218e2bd7c0b3d843a7cd1e30ef980430d44b51

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Some constructions of biharmonic maps and Chen's conjecture on biharmonic hypersurfaces (2009, background): “For example, one can check that the following family (in parameter s) of maps: φs : S1×S1 −→ S(3) defined by φs(x, y) = (cos s cosx, cos s sin x, sin s cos y, sin s sin y) are harmonic maps for all s ∈ [0, π/2] (see [2] for details).”
Slant submersions from almost Hermitian manifolds (2010, methodology): “on τ(ϕ) of Γ(ϕ−1TN) deﬁned by τ(ϕ) = divϕ∗ = Xm i=1 (∇ϕ∗)(ei,ei), (2.12) where {e 1,...,em} is the orthonormal frame on M. Then it follows that ϕ is harmonic if and only if τ(ϕ) = 0, for details, see [1] 3.Slant Submersions In this section, we deﬁne slant submersions from an almost Hermitian manifol”
Biharmonic hypersurfaces in Riemannian manifolds (2009, background): “, the second fundamental form ∇dφ(X, Y ) of the isometric immersion φ agrees with the second fundamental form B(X, Y ) of the immersed submanifold φ(M) in N (see [17], Chapter 7, also [2], Example 3.”
Conformal Riemannian Maps between Riemannian Manifolds, Their Harmonicity and Decomposition Theorems (2010, background): “We note that dim(M) < dim(N) for a nonconstant weakly conformal map [3].”
Slant submersions from almost product Riemannian manifolds (2013, neutral): “Recall that π is said to be harmonic if trace(∇π∗) = 0 and π is called a totally geodesic map if (∇π∗)(X,Y ) = 0 for X,Y ∈ Γ(TM) [2].”
Quaternionic Monge-Ampère equations (2002, background): “[ 9 ] for more details and references) . However there is a general result of B. Fuglede [21] which says the following.”
A short survey on biharmonic maps between Riemannian manifolds (2005, background): “In analogy with the case of harmonic morphisms (see [4]) the definition of biharmonic morphisms can be formulated as follows.”
Biharmonic maps into a Riemannian manifold of non-positive curvature (2012, neutral): “[1], [3]) by τ(φ) = n− 2 2 λ(2) dφ ( gradH ( 1 λ2 )) − (m− n)dφ ( Ĥ ) , (3.”
The stress-energy tensor for biharmonic maps (2006, neutral): “11 A submersion φ : (M, g) → (N , h) is horizontally conformal ([3]) if there exists a positive function λ ∈ C∞(M), the dilation, such that h(dφp(X), dφp(Y )) = λ(2)gp(X, Y ), ∀X, Y ∈ (ker dφp)⊥.”
Anti-invariant Riemannian submersions from almost Hermitian manifolds (2010, background): “Then it follows that φ is harmonic if and only if τ(φ) = 0, for details, see [2]”