### Citations

460 | H.: Harmonic mappings of Riemannian manifolds - Eells, Sampson - 1964 |

151 |
Complex Analytic Coordinates in Almost Complex Manifolds,
- Newlander, Nirenberg
- 1957
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Citation Context ... (4) by giving the dependent and independent variables, e.g. J(M) = J(V (M)) and J(µ) = J(V (M(µ))), (i.e. ι), and their inverse maps by M(J), µ(J) etc. For computations, the following is useful (cf. =-=[16]-=-): Lemma 2.2 The (0, 1)-tangent space of Jx has basis {e ī : i = 1, . . .,m} where eī = ∂ + Mjī ∂qī ∂ . ∂qj Proof It suffices to note that, under the natural pairing of T ⋆c x (R 2m ) and T c x (R2m )... |

122 |
Treatise on the Theory of Determinants,
- Muir
- 1960
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Citation Context ...r+s E 12 rs r,s=2 r<s ∣ (−1) r+s E 12 rs ∣ Kr 1 Ks 1 Kr 2 Ks 2 ∣ ⎫ ⎪⎬ ∣⎪⎭ . ∣ Kr 1 Ks 1 Kr 2 Ks 2 ⎫ ⎪⎬ ∣⎪⎭ 16That the expression in brackets is − detK follows from Laplace’s Theorem for determinants =-=[15]-=-. Proof of Claim 3.15 It suffices to prove the case r < s. We apply Sylvester’s Theorem [15]. For this we need some notation. Denote the (k − 2) × (k − 2)determinant obtained from K by omitting rows 1... |

106 |
Harmonic morphisms between riemannian manifolds.
- Fuglede
- 1978
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Citation Context .... . . x n ) ∈ U. (2) (Equivalently, and in more generality, a harmonic morphism is a map between Riemannian manifolds which pulls back germs of harmonic functions to germs of harmonic functions — see =-=[7, 12]-=-.) ∗ Partially supported by EC grant CHRX-CT92-0050 1Suppose that n is even, say n = 2m. Then it is well-known that a map φ : U → C which is holomorphic with respect to the standard complex structure... |

104 | A mapping of riemannian manifolds which preserves harmonic functions. - Ishihara - 1979 |

62 |
A conservation law for harmonic maps
- Baird, Eells
- 1981
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Citation Context ...to be submersive or not. (iii) Thus, for all n > 4, we have full globally defined harmonic morphisms R n → C which do not arise (see Definition 4.8) from Kähler structures (Corollary 4.21). Since, by =-=[2]-=-, the fibres of a submersive harmonic morphism to C (or to a Riemann surface) form a conformal foliation by minimal submanifolds of codimension 2, we obtain many interesting such foliations, in partic... |

41 |
Harmonic morphisms and Hermitian structures on Einstein 4-manifolds
- Wood
- 1992
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Citation Context ...monic if and only if the divergence of J lies in the kernel of the differential of φ (see, for example, [11]). It is not easy to see how to find maps satisfying this condition in general. However, in =-=[17]-=- with some further development in [1], strong relationships were shown between Hermitian structures and harmonic maps from R 4 , (or, more generally, any 4-dimensional anti-self-dual Einstein manifold... |

25 |
A nonlinear elliptic system for maps from Hermitian to Riemannian manifolds and rigidity theorems in Hermitian geometry
- Jost, Yau
- 1993
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Citation Context ... (11) below), a holomorphic map (U, J) → C still satisfies (2), however it is no longer automatically harmonic. (Note that it is, however, always a Hermitian harmonic map in the sense of Jost and Yau =-=[13]-=-.) In fact it is harmonic if and only if the divergence of J lies in the kernel of the differential of φ (see, for example, [11]). It is not easy to see how to find maps satisfying this condition in g... |

15 |
Harmonic morphisms from complex projective spaces, Geometriae Dedicata 53
- Gudmundsson
- 1994
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Citation Context ...say that φ reduces to A and ψ is a reduction of φ. Note that if φ does so factor, then it is a harmonic map (resp. harmonic morphism) if and only if ψ is a harmonic map (resp. harmonic morphism), see =-=[8]-=-. Similarly, we shall say that a foliation F on U is full if it is not the inverse image (πA) −1 (F ′ ) of a foliation F ′ on πA(U) for any subspace A. We now give a test for fullness for holomorphic ... |

11 |
Riemannian twistors and Hermitian structures on low-dimensional space forms
- Baird
- 1992
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Citation Context ... J lies in the kernel of the differential of φ (see, for example, [11]). It is not easy to see how to find maps satisfying this condition in general. However, in [17] with some further development in =-=[1]-=-, strong relationships were shown between Hermitian structures and harmonic maps from R 4 , (or, more generally, any 4-dimensional anti-self-dual Einstein manifold) to C or to a Riemann surface; in pa... |

6 |
Bartolomeis, Applications harmoniques stables dans
- Burns, de
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Citation Context ...structure is integrable. Next note that an almost complex structure J on an open subset U of R 2m can be considered as a map J : U → SO(2m)/U(m) or as a section σJ : U → Z + ; then we have: Lemma 2.1 =-=[5, 4]-=- Let J be a positive almost Hermitian structure on U open ⊂ R2m . Then the following are equivalent: 1. J is integrable; 2. the corresponding map J : U → SO(2m)/U(m) is holomorphic; 3. the correspondi... |

6 |
Non-holomorphic harmonic morphisms from Kähler manifolds
- Gudmundsson
- 1994
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Citation Context ...ian of such a composition f ◦ φ is given by ∆(f ◦ φ) = = 0 ∂ 2 f ∂zj∂z k 〈∇φj, ∇φ k 〉 c + ∂f ∂z l ∆φl 10where ∇φ j = (∂φ j /∂x 1 , . . . , ∂φ j /∂x 2m ) for (x 1 , . . . , x 2m ) ∈ R 2m . Now, as in =-=[9]-=-, ∇φ j , ∇φ k and ∇φ j + ∇φ k , being the gradients of holomorphic functions, are isotropic, so by the polarization identity, 〈∇φ j , ∇φ k 〉 c = 0. There is one special case when all holomorphic maps ... |

2 |
Harmonic morphisms and minimal submanifolds of spheres and projective spaces, preprint in preparation
- Baird, Wood
(Show Context)
Citation Context ...nµ1 = µ1(z 1 , z 2 ). This example is additionally interesting because z 1 is invariant under radial translation q ↦→ aq, 0 < a < ∞, and so reduces to a full harmonic morphism on a domain in S 7 (see =-=[3]-=-). Generalisation of these constructions to arbitrary m > 2 is straightforward. Global examples in even dimensions The above examples show the richness and variety that abounds for m > 2. We may ask w... |