Characterizations for Riemannian submersions to be harmonic or biharmonic are shown. Examples of biharmonic but not harmonic Riemannian submersions are shown.

2018-09-28math.DG

Harmonic Riemannian submersions from 3-dimensional geometries

In this paper, we study harmonic Riemannian submersions from 3-dimensional geometries using the ( generalized) integrability data associated to an orthonormal frame natural to a Riemannian submersion. We give complete classifications of harmonic Riemannian submersions from Thurston's 3-dimensional geometries, 3-dimensional BCV spaces and Berger sphere into a surface. We also give some explicit constructions of these harmonic Riemannian submersions.

2023-02-19math.DG

k-谐波浸入和浸入球体

k-Harmonic immersion and submersion into a sphere

J. Eells和L. Lemaire引入了k-调和映射，王少波给出了第一变分公式。当k=2时，称为双调和映射（2-调和映射）.在这一领域进行了广泛的研究。本文研究了球面中的k-调和浸入，得到了半径与k-调和的k之间的关系。我们还考虑了k-调和淹没，推广了Oniciuc的结果.然后利用Hopf映射构造了非调和k-调和函数。

2011-08-05math.DG

Harmonic and biharmonic Riemannain submersions from Sol space

In this paper, we give a complete classification of harmonic and biharmonic Riemannian submersions $π:(R^3,g_{Sol})\to (N^2,h)$ from Sol space into a surface by proving that there is neither harmonic nor biharmonic Riemannian submersion $π:(R^3,g_{Sol})\to (N^2,h)$ from Sol space no matter what the base space $(N^2,h)$ is. We also prove that a Riemannian submersion $π:(R,g_{Sol})\to (N^2,h)$ from Sol space exists only when the base space is a hyperbolic space form.

2023-02-19math.DG

调和几乎淹没的特征值

Eigenvalues of harmonic almost submersions

利用目标度量的拉回特征值，即第一基本型，研究了在稠密子集上浸没的黎曼流形之间的映射。这些特征值的导数的表达式给出了调和性、全测地映射和度量保持调和性的双共形变换的刻画。利用特征值的伸缩和在五维空间中的Bochner技巧方法，包括特征值差的Laplacian，证明了伪调和态射的Schwarz引理，给出了伪调和态射成为调和态射的条件.

2008-09-09math.DG

非紧型黎曼对称空间之间的调和黎曼淹没

Harmonic Riemannian submersions between Riemannian symmetric spaces of noncompact type

该研究构造了非紧型对称空间到其一秩全测地子空间的调和黎曼淹没，证明了分裂实半单李群对应黎曼对称空间上存在全局非常值复值调和态射，完成了Gudmundsson猜想的肯定证明。

2024-04-12math.DG

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谐波接触度量结构和浸没

Harmonic contact metric structures, and submersions

我们研究调和几乎切触结构的上下文中的接触度量流形，并进行分析时，这样的流形纤维在几乎厄米流形，如Boothby-Wang纤维化。两种类型的几乎接触度量翘曲的产品也进行了研究，其调和性的基础上或纤维的几乎厄米特结构。

2007-08-31math.DG

Bifurcation for the constant scalar curvature equation and harmonic Riemannian submersions

We study bifurcation for the constant scalar curvature equation along a one-parameter family of Riemannian metrics on the total space of a harmonic Riemannian submersion. We provide an existence theorem for bifurcation points and a criterion to see that the conformal factors corresponding to the bifurcated metrics must be indeed constant along the fibers. In the case of the canonical variation of a Riemannian submersion with totally geodesic fibers, we characterize discreteness of the set of all degeneracy points along the family and give a sufficient condition to guarantee that bifurcation ne…

2018-08-16math.DG

Some links between $F$-harmonicity, submersion and cohomology

By studying cohomology classes that are related with $p$-harmonic morphisms, $F$-harmonic maps, and $f$-harmonic maps, we extend several of our previous results on Riemannian submersions and $p$-harmonic morphisms to $F$-harmonic maps, and $f$-harmonic maps which are submersions.

2023-03-21math.ATmath.DG

Kähler流形的反全纯半不变量浸没

Anti-holomorphic semi-invariant submersions from Kählerian manifolds

研究了从Kähler流形到黎曼流形的反全纯半不变量浸入.我们证明了所有的分布，其中涉及的定义的沉没是可积的。我们还证明了O'Neill张量$\mathcal{T}$在不变垂直分布上为零。我们给出了这类浸没的完全地干性和调和性的充要条件。此外，我们研究了全流形和纤维的几种曲率，并给出了一个特征定理。

2014-04-09math.DG

Generalized harmonic morphisms and horizontally weakly conformal biharmonic maps

Harmonic morphisms are maps between Riemannian manifolds that pull back harmonic functions to harmonic functions. These maps are characterized as horizontally weakly conformal harmonic maps and they have many interesting links and applications to several areas in mathematics (see the book by Baird and Wood for details). In this paper, we study generalized harmonic morphisms which are defined to be maps between Riemannian manifolds that pull back harmonic functions to biharmonic functions. We obtain some characterizations of generalized harmonic morphisms into a Euclidean space and give two met…

2017-12-10math.DG

三维流形上的双调和黎曼浸入

Biharmonic Riemannian submersions from 3-manifolds

Chen-Ishikawa [CI]和Jiang [Ji]独立证明了关于双调和子流形的一个重要定理：曲面到三维欧几里得空间的等距浸入是双调和的当且仅当它是调和的（即极小的）。在后来的论文[CMO 2]中，Cadeo-Monttaldo-Oniciuc证明了如果目标欧几里得空间被三维双曲空间型所取代，则该定理仍然成立。本文证明了黎曼浸没的对偶结果，即，从非正曲率的三维空间型到曲面的黎曼浸入是双调和的当且仅当它是调和的。

2010-02-24math.APmath.DG