Sigmundur Gudmundsson, Marko Sobak
We introduce a new method for constructing complex-valued $r$-harmonic functions on Riemannian manifolds. We then apply this method for the important semisimple Lie groups $SO(n)$, $SU(n)$, $Sp(n)$, $SL_n(R)$, $Sp(R,n)$, $SU(p,q)$, $SO(p,q)$, $Sp(p,q)$, $SO^*(2n)$ and $SU^*(2n)$.
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Harmonic vector fields on pseudo-Riemannian manifolds
将黎曼流形上的调和向量场理论推广到伪黎曼流形上。伪欧几里德超二次曲面上的调和共形梯度场被分类为同余，伪黎曼二次曲面上的调和Killing场也是如此。利用仿Kaehler扭曲反等距映射对中性签名二次曲面上的调和向量场进行相关。
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.
On transversally harmonic maps of foliated Riemannian manifolds
研究了叶化黎曼流形之间的横截调和映射。特别地，我们证明了在一定的曲率条件下，横截调和映射是横截全测地的。
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。
Harmonic morphisms with one-dimensional fibres on conformally-flat Riemannian manifolds
我们将一维纤维调和态射分类为（1）至少四维的实解析共形平坦黎曼流形和（2）至少三维的共形平坦黎曼流形。
Harmonic maps into sub-Riemannian Lie groups
该研究推广黎曼流形的相关定义，提出次黎曼流形间调和映射的定义，推导了映到左不变度量李群的水平映射为调和映射的条件，证明其可分为异常与正常两类，并以Heisenberg群的调和映射方程为例进行说明。
We apply the method of eigenfamilies to construct new explicit complex-valued p-harmonic functions on the non-compact classical Lie groups, equipped with their natural semi-Riemannian metrics. We then employ this same approach to manufacture explicit complex-valued harmonic morphisms on these groups.
In this paper, we consider critical maps of a horizontal energy functional for maps from a sub-Riemannian manifold to a Riemannian manifold. These critical maps are referred to as subelliptic harmonic maps. In terms of the subelliptic harmonic map heat flow, we investigate the existence problem for subelliptic harmonic maps. Under the assumption that the target Riemannian manifold has non-positive sectional curvature, we prove some Eells-Sampson type existence results for this flow when the source manifold is either a step-2 sub-Riemannian manifold or a step-r sub-Riemannian manifold whose sub…
RC-positivity and rigidity of harmonic maps into Riemannian manifolds
本文证明了从具有一致RC-正曲率的紧致Kähler流形到具有非正复截面曲率的黎曼流形的每个调和映射都是常数.特别地，不存在从具有正全纯截面曲率的紧致Kähler流形到具有非正复截面曲率的黎曼流形的非常调和映射.
k-harmonic curves into a Riemannian manifold with constant sectional curvature
J.Eells和L. Lemaire引入了k-调和映射，T. Ichiyama，J. Inoguchi和H.Urakawa给出了一阶变分公式。本文将$3$ -调和曲线的常微分方程描述到具有常截面曲率的黎曼流形中，证明了双调和曲线是k-调和曲线$(k\geq 2)$ .
Vanishing theorems of the basic harmonic forms on a complete foliated Riemannian manifold
在具有某些横曲率条件的紧致叶状黎曼流形上，不存在非平凡的基本调和形式（M。Min-Oo等人，J. Reine Angew。Math.415（1991）。本文将上述事实推广到完备的叶状黎曼流形。
We classify biharmonic and harmonic homomorphisms $f:(G,g_1)\rightarrow(G,g_2)$ where $G$ is a connected and simply connected three-dimensional unimodular Lie group and $g_1$ and $g_2$ are left invariant Riemannian metrics.
Riemannian foliations with parallel or harmonic basic forms
本文考虑法丛具有平行或调和基本形式的黎曼叶理。我们估计的范数的O'Neill张量的曲率数据的整个流形。然后给出了一些例子。
Harmonic morphisms between degenerate semi-Riemannian manifolds
本文将调和映射和态射推广到\emph{degenerate semi-Riemannian category}，当流形$M$和$N$是\emph{stationary}，映射$ϕ:M\to N$是\emph{radical-preserving}时.我们几何刻画了\emph{(generalized) horizontal (weak) conformality}的概念，并得到了（广义）调和态射的（广义）调和映射的一个刻画。
Harmonic sections of Riemannian vector bundles, and metrics of Cheeger-Gromoll type
本文研究了黎曼向量丛的调和截面，其全空间具有一个包含Sasaki和Cheeger-Gromoll度量的两参数度量族.这使得调和单位截面理论可以推广到非零Euler类的丛。
Given two Riemannian manifolds $(B,g_B)$ and $(F,g_F)$, we give harmonicity conditions for vector fields on the Riemannian warped product $B\times_fF$, with $f:B \longrightarrow ]0,+\infty[$, using a characteristic variational condition. Then, we apply this to the case $B=\mathbb{R}$ and $F$ is a three-dimensional connected Riemannian Lie group $G$ equipped with a left-invariant metric, to determine harmonic vector fields on $\mathbb{R}\times_fG$. We give examples of harmonic vector fields on $G$ which are not left-invariant and determine harmonic vector fields on $\mathbb{R}\times_fG$. We con…
On f-harmonic morphisms between Riemannian manifolds
f-调和映射首先由Lichnerowicz在\cite{Li}中引入并研究（另见Eells-Lemaire的报告\cite{EL}的第10.20节）。本文研究了f-调和映射的一个子类f-调和态射，它将局部调和函数拉回到局部f-调和函数.证明了黎曼流形之间的映射是f-调和态射当且仅当它是水平弱共形f-调和映射。这推广了著名的Fuglede-Ishihara对调和态射的刻画。给出了f-调和态射的一些性质和例子，以及f-调和态射的不存在性。我们还研究了共形浸入的f-调和性。
We introduce the natural notion of (p,q)-harmonic morphisms between Riemannian manifolds. This unifies several theories that have been studied during the last decades. We then study the special case when the maps involved are complex-valued. For these we find a characterisation and provide new non-trivial examples in important cases.
On compact Riemannian manifolds with harmonic weyl curvature
给出了具有调和Weyl曲率、正数量曲率和正常数$σ_2$的n $(\geq4)$维紧致黎曼流形的刚性定理.当$n=4,$时，我们证明了具有正数量曲率和正常数$σ_2$的4维紧致局部共形平坦黎曼流形等距于圆$\mathbb{S}^4$的商.
Pseudo-Harmonic Maps From Pseudo-Hermitian Manifolds to Riemannian Manifolds
本文讨论了从闭伪Hermite流形到具有非正截面曲率的黎曼流形的伪调和映射的热流，证明了伪调和映射的存在性，它是Eells-Sampson存在性定理的推广.在目标流形具有负截面曲率的条件下，我们还讨论了它的同伦类的伪调和表示的唯一性，这是Hartman定理的推广。
We show that any left invariant metric with harmonic curvature on a solvable Lie group is Ricci-parallel. We show the same result for any Lie group of dimension $\leq$ 6.
Harmonic maps on weighted Riemannian foliations
本文研究加权叶状结构上的横截f-调和映射与$(F,F')_f$-调和映射，推广了极小叶状结构中两类调和映射的等价关系。
Back to harmonic mappings of compact Riemannian manifolds
本文探讨了黎曼流形间调和映射理论的几个关联问题，提出了验证调和性的解析准则，分析了拉回度量的L2-正交分解，并研究了调和对称双线性形式在恒等调和映射中的作用。
In this work we construct explicit complex-valued p-harmonic functions on the compact Riemannian symmetric spaces SU(n)/SO(n), Sp(n)/U(n), SO(2n)/U(n), SU(2n)/Sp(n). We also describe how the same can be manufactured on their non-compact symmetric dual spaces.
Vanishing theorems for $L^2$ harmonic forms on complete Riemannian manifolds
本文给出了完备黎曼流形上的$L^2$调和形式的几个消灭定理，它们具有加权Poincaré不等式和一定的曲率下界.这些结果继承了Li-Wang和Lam的思想，但没有对权函数的符号和增长率作任何假设，因此可以应用于完备稳定超曲面.
k-harmonic maps into a Riemannian manifold with constant sectional curvature
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。本文研究了重调和映射与k-调和映射之间的关系，给出了3-调和映射的不存在性定理。给出了欧几里得空间的k-调和子流形的定义，并研究了欧几里得空间中的k-调和曲线.其次，我们给出了欧几里得空间的k-调和子流形的一个猜想.
Vanishing properties of $p$-harmonic $\ell$-forms on Riemannian manifolds
本文给出了黎曼流形（$p\geq2$）上$p$ -调和$\ell$ -形式的几个消失型定理.首先，我们考虑${N}^{n+m}$的完备非紧浸入子流形$M^n$，它具有平坦的法丛，证明了$M$上的任何$p$ -调和$\ell$ -形式是平凡的，如果$N$具有纯曲率张量，并且$M$满足一定的几何条件.然后，我们得到了黎曼流形上加权Poincaré不等式的一个消失定理.最后，我们研究了完备单连通局部共形平坦的黎曼流形$M$，指出只要$\operatorname{Ric}$有适当的界，$M$上就不存在非平凡的$p$ -调和$\ell$ -形式.
In this paper we consider the local $L^p$ estimate of Riemannian curvature for the Ricci-harmonic flow or List's flow introduced by List \cite{List2005} on complete noncompact manifolds. As an application, under the assumption that the flow exists on a finite time interval $[0,T)$ and the Ricci curvature is uniformly bounded, we prove that the $L^p$ norm of Riemannian curvature is bounded, and then, applying the De Giorgi-Nash-Moser iteration method, obtain the local boundedness of Riemannian curvature and consequently the flow can be continuously extended past $T$.
Liouville Theorem for Harmonic Maps from Riemannian Manifold with Compact Boundary
本文为从带紧致边界的完备非紧黎曼流形（Kasue流形）到非正截面曲率单连通完备黎曼流形的调和映射提供了梯度估计，并由此得出刘维尔定理及线性椭圆方程无正解的结论。
We find geometric conditions on a four-dimensional Hermitian manifold endowed with a metric connection with totally skew-symmetric torsion under which the complex structure is a harmonic map from the manifold into its twistor space considered with a natural family of Riemannian metrics defined by means of the metric and the given connection on the base manifold.
Harmonic sections of tangent bundles equipped with Riemannian $g$-natural metrics
设$(M,g)$是一个黎曼流形.当$M$是紧的且切丛$TM$具有Sasaki度量$g^s$时，定义从$(M,g)$到$(TM,g^s)$的调和映射的向量场是唯一的平行向量场. Sasaki度量和其他熟知的关于$TM$的黎曼度量是$g$ -自然度量的特定示例。我们给$TM$配上一个任意的黎曼$g$ -自然度规$G$，并研究了$M$的向量场$V$的调和性，认为它是从$(M,g)$到$(TM,G)$的映射.然后，我们将这项研究应用到Reeb向量场，特别是奇维球面上的Hopf向量场。
In the previous paper [GLM2018], we showed that the theory of harmonic maps between Riemannian manifolds may be discretized by introducing triangulations with vertex and edge weights on the domain manifold. In the present paper, we study convergence of the discrete theory to the smooth theory when taking finer and finer triangulations. We present suitable conditions on the weighted triangulations that ensure convergence of discrete harmonic maps to smooth harmonic maps. Our computer software Harmony implements these methods to computes equivariant harmonic maps in the hyperbolic plane.
Local Gradient Estimate for $p$-harmonic functions on Riemannian Manifolds
对于黎曼流形上的正$p$ -调和函数，我们得到了一个梯度估计和Harnack不等式，其常数仅依赖于里奇曲率的下界、$n$维数、$p$维数和函数所定义的球的半径.我们的方法是基于一个仔细的应用Moser迭代技术和不同的Cheng-Yau的方法所采用的Kostchwar和Ni，其中一个梯度估计的正$p$ -调和函数的截面曲率是从下面有界的假设下导出。
Harmonic Mappings into non-negatively curved Riemannian manifolds
50年前，Eells和Sampson证明了一个著名的定理：如果调和映射$(M, g)$是一个具有非负Ricci张量的紧致流形，且$(\bar{M},\bar{g})$的截面曲率是非正的，则调和映射是全测地的.此外，调和映射理论的其他主要结果“在大”是调和映射到非正弯曲的黎曼流形的结果。本文发展了到具有非负截面曲率的黎曼流形的调和映射理论。特别地，我们将证明，如果$(\bar{M},\bar{g})$的截面曲率是非负的，并且$(M, g)$是一个紧流形，其Ricci张量$Ric \geq f^{*}\bar{Ric}$是由$f$拉回的Ricci张量$f^{*}\bar{Ric}$，则任何黎曼流形$\bar{Ric}$之间的调和映射都是全测地的。将上述方案推广到完备流形到具有非负截面曲率流形的调和映射。此外，我们将从我们的结果中得到有趣的推论。
Harmonic maps and biharmonic Riemanian submersions
给出了黎曼浸没是调和的或双调和的刻划。双调和但不是调和黎曼浸没的例子。
Integral Liouville theorem in a complete Riemannian manifold
如果黎曼流形中的Killing向量场是光滑实值函数的梯度，则称之为Killing势。本文给出了完备黎曼流形中Killing势存在的一个必要条件。Yau利用梯度估计证明了黎曼流形上调和函数的Liouville定理，此后许多作者推广了这一概念，并研究了各种类型的调和函数的Liouville定理。本文还证明了完备黎曼流形上调和函数积分形式的Liouville定理。最后研究了满足梯度Ricci孤立子的完备黎曼流形中调和函数的性质，证明了调和函数是沿测地线的距离函数的常数倍。
4-dimensional Riemannian manifolds with a harmonic 2-form of constant length
这是表明，如果一个紧凑的，有向的4维黎曼流形（M，g）的正截面曲率承认调和2形式的常数长度，它的交叉形式是明确的，这样的调和形式是唯一的常数倍。在本文中，我们证明了这样一个流形在较弱的曲率假设下与$\mathbb{CP}_{2}$同构，并且在$\mathbb{CP}_{2}$上的Fubini-Study度量附近存在这样一个度量的无限维模空间.我们讨论了一些条件，可以添加，以获得Fubini-Study度量的复同态和重标度。
The variational theory of higher-power energy is developed for mappings between Riemannian manifolds, and more generally sections of submersions of Riemannian manifolds, and applied to sections of Riemannian vector bundles and their sphere subbundles. A complete classification is then given for left-invariant vector fields on 3-dimensional unimodular Lie groups equipped with an arbitrary left-invariant Riemannian metric.
On $f$-bi-harmonic maps between Riemannian manifolds
双调和映射和$f$ -调和映射都有很好的物理基础和应用。本文通过这两种调和映射的结合，引入并研究了$f$ -双调和映射作为$f$ -双能泛函$\frac{1}{2}\int_M f|τ(ϕ)|^2dv_{g}$的临界点。这类映射推广了调和映射和双调和映射的概念。首先导出了$f$ -双调和映射方程，然后利用它研究了一些特殊映射的$f$ -双调和性，包括同维流形之间的共形映射，直积流形与单翘曲积流形之间的乘积映射，翘曲积流形上的投影映射和包含映射.
Bifurcation for the constant scalar curvature equation and harmonic Riemannian submersions
在调和黎曼浸没的全空间上研究了常数量曲率方程沿单参数黎曼度量族的分支.我们提供了一个存在定理的歧点和一个标准，看到的共形因子对应的分歧度量必须确实是常数沿纤维。在全测地纤维黎曼浸入的正则变分情形下，我们刻画了该族所有退化点的离散性，并给出了一个充分条件，保证线性化方程在每一个有非平凡解的点处都必然发生分支.在四元数Hopf纤维化模型的情况下，我们表明，除了在轮度量，不发生的破缺分岔。
Harmonic Gauge on the Space of Riemannian Metrics and Its Role in the Ricci-DeTurck Flow
本文研究了黎曼度量空间上的调和度量，证明其在Hilbert-Einstein函数的变分中消除散度项，并诱导出自然的椭圆结构，从而建立 gauge fixing、椭圆算子理论与几何刚性的联系。
Harmonic diffeomorphisms between pseudo-riemannian surfaces
本文在统一形式化框架下研究黎曼曲面与洛伦兹曲面间的调和映射，通过广义 sine-Gordon 方程分类并构造单孤子解，并利用 Bäcklund 变换建立不同方程解之间的联系。
Minimal graphs over Riemannian surfaces and harmonic diffeomorphisms
在有限拓扑的曲率为$-1$且面积为无穷大的完备黎曼曲面$Σ$上构造了一个抛物型的整极小图$S$。该图的垂直投影产生从$S$到$Σ$的调和同构。证明使用发散线理论来构造极小图。我们还推广了R.肖恩设$g_1$和$g_2$是具有紧边界的可定向曲面$S$上的两个完备度量，并对某些$C>0$和所有$r>0$设$$\int_{S_r^2}K_{g_2}^-dσ_{g_2}\le C\ln(2+r)$$.如果存在从$(S,g_1)$到$(S,g_2)$的调和同态，则$(S,g_1)$是抛物线。
In this paper, we give an estimate of sub-Laplacian of Riemannian distance functions in pseudo-Hermitian geometry which plays a similar role as Laplacian comparison theorem in Riemannian geometry, and deduce a prior horizontal gradient estimate of pseudo-harmonic maps from pseudo-Hermitian manifolds to regular balls of Riemannian manifolds. As an application, Liouville theorem is established under the conditions of nonnegative pseudo-Hermitian Ricci curvature and vanishing pseudo-Hermitian torsion. Moreover, we obtain the existence of pseudo-harmonic maps from complete noncompact pseudo-Hermit…
The theory of harmonic symmetric bilinear forms on a Riemannian manifold is an analogue of the theory of harmonic exterior differential forms on this manifold. To show this, we must consider every symmetric bilinear form on a Riemannian manifold as a one-form with values in the cotangent bundle of this manifold. In this case, there are the exterior differential and codifferential defined on the vector space of these differential one-forms. Then a symmetric bilinear form is said to be harmonic if it is closed and coclosed as a one-form with values in the cotangent bundle of a Riemannian manifol…
Let $(M,H,g_H;g)$ be a sub-Riemannian manifold and $(N,h)$ be a Riemannian manifold. For a smooth map $u: M \to N$, we consider the energy functional $E_G(u) = \frac{1}{2} \int_M[|\mathrm{d}u_H|^2-2G(u)] \mathrm{d}V_M$, where $\mathrm{d}u_H$ is the horizontal differential of $u$, $G:N\to \mathbb{R}$ is a smooth function on $N$. The critical maps of $E_G(u)$ are referred to as subelliptic harmonic maps with potential $G$. In this paper, we investigate the existence problem for subelliptic harmonic maps with potentials by a subelliptic heat flow. Assuming that the target Riemannian manifold has …
Reflection Principles for Smooth Harmonic Maps between Riemannian Manifolds and a Schwarz Reflection Principle for Harmonic and Holomorphic Maps from a Class of Hermitian Symmetric Spaces
在建立了两个黎曼流形M和N之间调和映射的局部Cauchy数据延拓的唯一性之后，我们证明了（i）包含M的反射子流形作为超曲面的黎曼流形M的光滑极小子流形Y的反射原理和（ii）包含反射子流形的一类黎曼流形之间的光滑调和映射的反射原理.从上述反射原理可以很容易地得出M上的调和函数关于M的反射超曲面的反射原理。为了研究一般厄米对称空间M上调和映射和全纯映射的反射原理，引入了递归实型的概念。厄米特对称空间包括常全纯曲率的厄米特流形、复超二次曲面和它们各自的非紧空间。设M是复维n的不可约Hermite对称空间，具有递归实型B1和相应的对合等距s1，X是光滑黎曼流形，B2是反射子流形，相应的对合等距s2 .又设h是从M到X的调和映射，使得（i）h（B1）在B2中，（ii）h将垂直于B1的切向量映射到垂直于B2的向量。我们将证明在M中h（s1（p））= s2（h（p））所有p。上述断言的一个特例是X是Hermite对称空间，B_2是X的实形式，h是全纯映射。这两个结论将经典的关于复平面上调和函数和全纯函数的Schwartz反射原理推广到一类具有递归实型的Hermitian对称空间的调和映射和全纯映射。证明了定义在复平面上且在实直线上取实值的亚纯函数的反射原理。
Hermitian Harmonic maps and non-degenerate curvatures
本文研究了从Hermitian流形到Kaehler流形、Hermitian流形和黎曼流形的各种调和映射的存在性。利用厄米特空间上的精化Bochner公式，（可能非Kaehler）流形，得到了从紧致Hermitian流形到黎曼流形的Hermitian调和映射的新的刚性结果，同时也得到了从紧致复流形到紧致Kaehler流形的多重调和映射的复解析性（和黎曼流形），它们类似于[28，Siu]，[14，Jost-Yau]和[26，Sampson]中的几个基本结果.
Harmonic morphisms of sub-Riemannian Lie groups
本文证明次黎曼李群间的调和态射是光滑共形淹没且为拉普拉斯算子的对称性，给出李群外调和态射的部分结果，计算其水平能量第一变分，发现其不一定是调和映射。
Removability of singularities of harmonic maps into pseudo-Riemannian manifolds
本文考虑到伪黎曼流形的调和映射。本文证明了连续映射的孤立奇点的可移性，即从R^m的开子集到伪黎曼流形的任何连续映射，只要该流形是二次连续可微的，且在孤立点外处处调和，则它实际上是处处光滑调和的.
Yau-type gradient estimates for p-harmonic functions on Riemannian manifolds with boundary under Dirichlet boundary condition
本文在带边黎曼流形上，利用边界径向障碍与内部最大值原理，将Yau型梯度估计推广至p-调和函数，并导出Liouville定理。
On the behaviour of harmonic functions on Riemannian cones
本文探讨了黎曼锥上调和函数的性质，并研究了Liouville定理在该情境下的成立情况。
A Framework for Gluing Harmonic Maps
本文提出一个粘合框架，在特定条件下构造从黎曼曲面到黎曼流形 $N$ 的扩张调和映射的粘合映射。
A class of nonsymmetric harmonic Riemannian spaces
$H$型群的某些可解扩张为所谓的Lichnerowicz猜想提供了非紧的反例，该猜想断言“调和”黎曼空间必须是秩1对称空间。
In this paper, we investigate subelliptic harmonic maps with potential from noncompact complete sub-Riemannian manifolds corresponding to totally geodesic Riemannian foliations. Under some suitable conditions, we give the gradient estimates of these maps and establish a Liouville type result.
Unique continuation properties for polyharmonic maps between Riemannian manifolds
本文研究黎曼流形间k阶多重调和映射的唯一延拓原理，在连通定义域假设下证明了局部调和、局部重合及局部映入赤道均可延拓至全局。
On critical maps of the horizontal energy functional between Riemannian foliations
本文研究 Riemannian foliations 间光滑映射的水平能量泛函临界点，利用应力-能量张量建立单调性公式，并给出无穷远渐近条件下的 Jin 型定理。
In this paper, we mainly derive monotonicity formula of generalized map using conservation law, including $ϕ$-$F$ harmonic map coupled with $ϕ$-$F$ symphonic map with $m$ form and potential from metric measure space, $ p $ harmonic map with potential , $ V $ harmonic map with potential. As an corollary, we can derive Liouville theorem for these maps under some finite energy conditons. We also get Liouville type theorem for $ϕ$-$F$ harmonic map coupled with $ϕ$-$F$ symphonic map under asymptotic conditon on metric measure space. We also get Liouville theorem for $ϕ$-$F$-$V$-harmonic maps in ter…
Holomorphic harmonic morphisms from four-dimensional non-Einstein manifolds
我们构造了具有余维为2的最小叶的左不变共形叶理的四维黎曼李群。我们证明了这些叶理关于一个非K\”ahler的（可积）Hermite结构是全纯的.证明了所构造的黎曼李群是{ \it not} Einstein流形。这回答了黎曼四维流形复值调和态射理论中的一个重要的公开问题。
Generalized harmonic morphisms and horizontally weakly conformal biharmonic maps
调和态射是将调和函数拉回到调和函数的黎曼流形之间的映射。这些地图的特点是水平弱共形调和映射，他们有许多有趣的联系和应用的几个领域的数学（见书贝尔德和木材的细节）。本文研究了广义调和态射，广义调和态射定义为黎曼流形之间将调和函数拉回到双调和函数的映射。我们得到了欧几里得空间中广义调和态射的一些特征，并给出了两种构造方法，这些方法可以用来产生许多不是调和态射的广义调和态射的例子.给出了翘曲积空间投影间广义调和态射的完全分类，从而提供了翘曲积流形上真双调和黎曼浸入和共形浸入的无穷多个例子.
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