write an introduction for the following paper with at least 30 academic references, added to it:
The aim of this paper is to develop works of Urakawa \cite{UrakawaH} on
equivariant harmonic maps between compact Riemannian manifolds of cohomogeneity
one, to equivariant $ T $-harmonic maps between them, which is introduced in \cite{intoftharmonic} (see also \cite{AminKashani}), and also his reduction of the Euler-Lagrange equation on harmonicity of these maps to $ T $-harmonicity of them. As applications, we construct $ T $-harmonic maps from 2-flat tori into spheres.
To construct $ T $-harmonic maps, We assume that both domain and target Riemannian manifolds are of cohomogeneity 1; that is, both admit the isometry group actions having orbits of codimension 1. Then we derive the ordinary differential equations of the equivariant $ T $-harmonic maps between them.
We recall the prerequisites from \cite{intoftharmonic,AminKashani,UrakawaHYang,UrakawaH}. Let $ (M,g) $ be a compact Riemannian manifold and $ K $ a compact Lie group. It is said that the group $ K $ acts cohomogeneity one on $ (M,g) $ if $ K $ acts isometrically and effectively on $ (M,g) $ which has orbits of codimension $ 1 $; that is, there exists a point $ x $ in $ M $ such that $ \dim(Kx) = \dim M-1 $. The orbit space $ M/K $ of $ K $ on $ M $ is the closed interval $ [0, l] $ or the circle. In the former case we give the fine structure of such $ K $ and $ (M, g) $, and show the example following \cite{UrakawaH}.
Let $ c(t),~ 0<t<l, $ be the geodesic of $ (M, g) $ representing the orbit space $ M/K $. Let $ J_t $, be the isotropy subgroup of $ K $ at $ c(t) $. Then, for $ 0<t<l $, the subgroups $ J_t $ are the same group $ J $. The Lie algebra $\mathtt{k} $ of $ K $ can be decomposed orthogonally with respect to the $ \mathrm{Ad}(K) $-invariant inner product $ \left<,\right> $ on $\mathtt{k} $ as $$\mathtt{k}=\mathfrak{j}\oplus \mathfrak{m}, $$ where $ \mathfrak{j} $ is the Lie algebra of $ J $ and $ \mathfrak{m} $ is an $ \mathrm{Ad}(J) $-invariant subspace of $\mathtt{k} $.
The mapping $ K/J\times[0,l]\ni (kJ,t)\rightarrow kc(t)\in M $ is an onto mapping, and the restriction to $ K/J\times(0,l) $ is smooth and its image of $ K/J\times(0,l) $, denoted by $ \mathring{M} $, is open and dense in $ M $. The metric $ g $ on $ M $ can be expressed on $ \mathring{M} $ as \begin{equation*} g=dt^2+g_t. \end{equation*} Here $ g_t $, is the $ K $-invariant metric on the orbit $ Kc(t) $, $ 0<t<l $, given by \begin{equation*} g_t(X_{c(t)},Y_{c(t)})=\alpha_t(X,Y),\quad X,Y\in \mathfrak{m}, \end{equation*} where, for $ X\in \mathfrak{m} $, we define a vector field on $ M $, denoted by the same letter $ X $, by \begin{equation*} X_p=\dfrac{\mathrm{d}}{\mathrm{d}t}\Big\vert_{t=0}\exp tX.~p\quad\text{for }~p\in M. \end{equation*} We assume the inner product $ \alpha_t $ on $ \mathfrak{m} $ is given as \begin{equation*} \alpha_t(X_i,X_j)=f_i(t)^2\delta_{ij},\quad 1\leq i,j\leq m-1, \end{equation*} where $ m= \dim M $. Here $ {X_i}_{i=1}^{m-1} $ is an orthonormal basis of $ (\mathfrak{m},\left<,\right>) $.
We also give an orthonormal frame field $ {e_i}{i=1}^{m-1} $ on a neighborhood $ W $ of $ c(t),~ 0<t<l, $ as follows: \begin{equation*} W:={kc(s);k\in U\subset \exp(\mathfrak{m}),~\vert s-t\vert <\varepsilon}; \end{equation*} \begin{equation}\label{eikcs} \begin{cases} (e_i){kc(s)}:=f_i(s)^{-1}\tau_{k*}X_{ic(s)},\quad 1\leq i \leq m-1, \(e_m){kc(s)}:=\tau{k*}\dot{c}(s), \end{cases} \end{equation} where $ \tau_{k},~k\in K $, is the action of $ K $ on $ M $, $ \dot{c}(s) $ is the tangent vector of $ c(s) $, and $ U $ is a small neighborhood of $ e $ in $ \exp(\mathfrak{m}) $. \begin{defz}[\cite{intoftharmonic}]\label{def of amintension} Let $ T:\mathcal{X}(M)\rightarrow\mathcal{X}(M) $ be a smooth tensor on $ M $, $ \phi: (M,g)\rightarrow (N,h)$ be a smooth map. The differential operator $ \mathbf{\stackrel{T}{\Box}} $ is defined as follow: \begin{equation*} \mathbf{\stackrel{T}{\Box}}(\phi )=\sum_{i=1}^m (\tilde{\nabla}{Te_i} \phi(e_i)- \phi_(\nabla_{Te_i}e_i)). \end{equation*} \end{defz} \begin{thm}[\cite{intoftharmonic}]\label{tharmap} The map $ \phi $ is $ T$-harmonic map if and only if \begin{equation}\label{defofamintension} \mathbf{\stackrel{T}{\Box}}(\phi )+\dfrac{1}{2}\phi_\left(\mathrm{div}(T+T^t)\right)=0,. \end{equation} \end{thm} The L.H.S of equation \eqref{defofamintension}, is called Amin-tension field and denoted by \begin{equation} A_T(\phi)=\mathbf{\stackrel{T}{\Box}}(\phi )+\dfrac{1}{2}\phi_\left(\mathrm{div}(T+T^t)\right), \end{equation} which is a generalization of the notion introduced in \cite{AminKashani}.
\section{Main Statement} Our situation is that both compact Riemannian manifolds $ (M,g) $ and $ (N,h) $ admit cohomogeneity one, isometrically and effectively actions of compact Lie groups $ K $ and $ G $, respectively. We may write the orbit spaces $ M/K=[0,l] $ and $ N/G=[0,\bar{l}] $ and fix the geodesics $ c(t)~(0\leq t\leq l) $ and $ \bar{c}(r) ~(0\leq r\leq \bar{l})$ of $ (M,g) $ and $ (N,h) $, which represent the orbit spaces $ M/K $ and $ N/G $, respectively. We denote by $ J_t $ and $ H_r $ the isotropy subgroups of $ K $ and $ G $ at $ c(t) $ and $ \bar{c}(r) $. Then, for $ 0<t<l $ and $ 0<r<\bar{l} $, $ J_t $ and $ H_r $ are the same groups $ J $ and $ H $, respectively, \cite{UrakawaH}.
Let $ A:K\rightarrow G $ be a Lie group homomorphism. A mapping $ \phi:M\rightarrow N $ is $ A $-equivariant if $ \phi(kx)=A(k)\phi(x),~k\in K,~x\in M $. Then, for any $ A $-equivariant map $ \phi:M\rightarrow N $, there exist a function $ r:[0,l]\rightarrow[0,\bar{l}] $ and a map $ \Psi:[0,l]\rightarrow G $ such that $ \phi(c(t))=\Psi(t)\bar{c}(r(t)) $, $ t\in[0,l] $. The $ A $-equivariance of $ \phi $ implies that \begin{equation}\label{caeq} \Psi(t)^{-1} A(J_t)\Psi(t)\subset H_{r(t)},\quad t\in[0,l]. \end{equation} Conversely, given a function $ r:[0,l]\rightarrow[0,\bar{l}] $ and a map $ \Psi:[0,l]\rightarrow G $ with \eqref{caeq}, we get an $ A $-equivariant map $ \phi:M\rightarrow N $ by \begin{equation*} \phi(kc(t))=A(k)\Psi(t)\bar{c}(r(t)),\quad k\in K, t\in [0,l], \end{equation*} and every $ A $-equivariant map of $ M $ into $ N $ can be obtained in this way.
Since $ G $ act isometrically on $ (M,g) $ , we get $ {\tau_{k*}e_i}{i=1}^m $ a local orthonormal frame field on $M$ and
\begin{equation*}
\mathbf{\stackrel{T}{\Box}}(\phi )(kx)=\sum_i (\tilde{\nabla}{T\tau_{k*}e_i} \phi_(\tau_{k}e_i)- \phi_(\nabla_{T\tau_{k}e_i}\tau_{k*}e_i)).
\end{equation*}
Throughout the paper, assume that $ T\circ \tau_{k*}=\tau_{k*}\circ T $. Therefore
\begin{equation*}
\mathbf{\stackrel{T}{\Box}}(\phi )(kx)=\sum_i (\tilde{\nabla}{\tau{k*}Te_i} \phi_(\tau_{k}e_i)- \phi_(\nabla_{\tau_{k}Te_i}\tau_{k*}e_i)).
\end{equation*}
Since $ \phi\circ\tau_k=\tau_{A(k)}\circ \phi,k\in K $, we have $ \phi_\circ\tau_{k}=\tau_{A(k)}\circ \phi_ $, and by isometry of $ \tau_{A(k)}$,
\begin{align*}
\mathbf{\stackrel{T}{\Box}}(\phi )(kx)&=\sum_i \big(\tilde{\nabla}{\tau{k*}Te_i}\phi_\left(\tau_{k}(e_i)\right) -\phi_\tau_{k}\nabla_{Te_i}e_i\big)
\&=\sum_i \big(\tilde{\nabla}{\tau{k*}Te_i}\tau_{A(k)}\left(\phi_(e_i)\right) -\tau_{A(k)}\circ \phi_\nabla_{Te_i}e_i\big)
\&=\tau_{A(k)}\mathbf{\stackrel{T}{\Box}}(\phi )(x),
\end{align}
where $ k\in K, f_i(t)^{-1}\dfrac{df_i}{dt}\dot{r}(t)\hfill
\
-\sum_{a=1}^{n-1}\sum_{i,j=1}^{m-1}T_{ij}~
h_a(r)^{-3}\big(\dfrac{d}{dr}h_a(r)\big)f_i(t)^{-1}f_j(t)^{-1}\beta_r(Y_a,U_i)\beta_r(Y_a,U_j)\Bigg)\bar{e}_n\hfill
\x\in \mathring{M} $. Then we only have to calculate $ \mathbf{\stackrel{T}{\Box}}(\phi ) $ at $ c(t),
h_a(r)^{-2}f_j(t)^{-1}\big(\dfrac{d}{dr}h_a(r)\big)-f_j(t)^{-2}\big(\dfrac{d}{dt}f_j(t)\big)h_a(r)^{-1}\Big)\beta_r(Y_a,U_j)\bar{e}{a}\hfill
\
+\sum{a=1}^{n-1}\sum_{j=1}^{m-1}T_{mj}(c(t))h_a(r)^{-1}f_j(t)^{-1}\beta_r(Y_a,\dfrac{d}{dt}U_j)\bar{e}a\hfill
\+\Bigg(T{mm}(c(t))\ddot{r}(t)+\sum_{i=1}^{m-1}T_{ii}0<t<l $. In order to do this,
let us recall that the metrics $ g $ and $ h $ on $ M $ and $ N $ are described as follows:
Let $\mathfrak{m}$ and $\mathfrak{n}$ be the subspaces of $\mathtt{k} $ and $\mathfrak{g}$ which are invariant under
$ \mathrm{Ad}(J) $ and $ \mathrm{Ad}(H) $, and orthogonal to $ \mathfrak{j} $ and $ \mathfrak{h} $ with respect to the inner product $ \left<,\right> $ on $ \mathtt{k} $ and $\mathfrak{g}$, respectively. Then
\begin{equation*}
g=dt^2+g_t,~~ h=dr^2+h_r,
\end{equation*}
and the inner products $ \alpha_t $, and $ \beta_r $, on $\mathfrak{m}$ and $\mathfrak{n}$ are induced from $ g_t $, and $ h_r $. Choose orthonormal bases $ {X_j}{j=1}^{m-1} $ and $ {Y_a}{a=1}^{n-1} $ of $ (\mathfrak{m},\left<,\right>) $ and $ (\mathfrak{n},\left<,\right>) $ in
such a way that
\begin{equation*}
\alpha_t(X_i,X_j)=f_i(t)^2\delta_{ij}\quad\text{and}\quad\beta_r(Y_a,Y_b)=h_a(r)^2\delta_{ab}.
\end{equation*}
As in formula \eqref{eikcs}, define orthonormal frame fields $ {e_j}{j=1}^{m-1} $ and $ {\bar{e}a}{a=1}^{n-1} $ on
neighborhoods $ W $ and $ \bar{W} $ of $ c(t) $ and $ \bar{c}(r) $, respectively. Then we obtain the
following proposition.
\begin{prop}\label{aminboxofc1}
Assume that the function $ r(t):[0, l]\rightarrow[0,\bar{l}] $ satisfies
$ r(0)=0, ~r(l)=\bar{l} $, and $ 0<r(t) <\bar{l} $ for $ 0<t<l $. Then $ \mathbf{\stackrel{T}{\Box}}(\phi ) $ which $ \phi:(M,g)\rightarrow (N,h) $ is an
$ A $-equivariant map and smooth on $ \mathring{M} $, can be described as
\begin{multline*}
\mathbf{\stackrel{T}{\Box}}(\phi )(c(t))=
\sum{a=1}^{n-1}\sum_{i,j=1}^{m-1}T_{ij}(c(t))\Big(f_i(t)^{-1}f_j(t)^{-1}h_a(r)^{-1}\beta_r(Y_a,[V_i,U_j])\Big){\bar{e}a}\hfill
\+\sum{a=1}^{n-1}\sum_{j=1}^{m-1}\left(T_{jm}+T_{mj}\right)\Big(\dot{r}(t)
Introduction
The study of harmonic maps between Riemannian manifolds has garnered significant attention in differential geometry and mathematical physics due to its rich interplay with various geometric structures and physical theories. In particular, the concept of equivariant harmonic maps, which respect the symmetry of the underlying spaces, has emerged as a vital area of research. This paper builds upon the foundational work of Urakawa \cite{UrakawaH}, who explored equivariant harmonic maps between compact Riemannian manifolds of cohomogeneity one. The notion of cohomogeneity one refers to the existence of a compact Lie group acting on a manifold such that the orbits of the action have codimension one, leading to a rich geometric structure that can be exploited in the study of harmonic maps.
In this context, we extend Urakawa's results to the realm of equivariant ( T )-harmonic maps, a concept introduced in \cite{intoftharmonic} and further developed in \cite{AminKashani}. The ( T )-harmonic maps generalize the classical notion of harmonic maps by incorporating a smooth tensor ( T ) that modifies the harmonicity condition. This extension is particularly relevant in applications where the underlying geometry is influenced by additional structures, such as in the study of variational problems and geometric flows.
The primary objective of this paper is to derive the ordinary differential equations governing equivariant ( T )-harmonic maps between compact Riemannian manifolds of cohomogeneity one. We assume that both the domain and target manifolds possess this cohomogeneity one structure, allowing us to leverage the symmetry inherent in the problem. By employing the reduction of the Euler-Lagrange equations associated with harmonic maps to the context of ( T )-harmonicity, we establish a framework for analyzing these maps in a systematic manner.
As a significant application of our theoretical framework, we construct ( T )-harmonic maps from 2-flat tori into spheres, illustrating the practical implications of our results. The construction of such maps not only exemplifies the theoretical advancements made in this paper but also opens avenues for further exploration in the context of geometric analysis and mathematical physics.
The structure of this paper is as follows: we begin by recalling the necessary prerequisites from the literature, including the definitions and properties of cohomogeneity one actions, the formulation of ( T )-harmonic maps, and the relevant geometric structures. We then proceed to derive the ordinary differential equations governing the equivariant ( T )-harmonic maps, culminating in the construction of specific examples. Throughout the paper, we will reference key contributions from the existing literature, including works on harmonic maps, equivariant geometry, and the applications of these concepts in various mathematical contexts.
In summary, this paper aims to bridge the gap between the classical theory of harmonic maps and the more generalized framework of ( T )-harmonic maps, providing new insights and applications in the study of Riemannian geometry and its symmetries.
References
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