write an introduction for the following paper with at least 30 real 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)
The study of harmonic maps between Riemannian manifolds has been a significant area of research in differential geometry, with applications ranging from mathematical physics to the theory of minimal surfaces. In particular, the concept of equivariant harmonic maps, which respect the symmetry of the underlying spaces, has garnered attention due to its rich geometric structure and potential applications. This paper builds upon the foundational work of Urakawa \cite{UrakawaH}, who explored equivariant harmonic maps between compact Riemannian manifolds of cohomogeneity one. Our objective is to extend Urakawa's results to the realm of equivariant ( T )-harmonic maps, a concept introduced in \cite{intoftharmonic} and further developed in \cite{AminKashani}.
Equivariant harmonic maps are defined in the context of Riemannian manifolds that admit isometric actions by compact Lie groups, leading to a natural decomposition of the manifold into orbits of lower dimension. Specifically, we focus on manifolds of cohomogeneity one, where the action of the group results in orbits of codimension one. This setting allows for a detailed analysis of the geometric properties of the maps involved, as well as the associated Euler-Lagrange equations governing their harmonicity. The reduction of these equations to the ( T )-harmonicity condition, as established by Urakawa, provides a powerful framework for understanding the behavior of these maps under the influence of additional geometric structures.
In this paper, we will derive the ordinary differential equations governing equivariant ( T )-harmonic maps between compact Riemannian manifolds of cohomogeneity one. We will also present applications of our findings, including the construction of ( T )-harmonic maps from 2-flat tori into spheres. The results will be framed within the context of existing literature, highlighting the contributions of various authors to the field of harmonic maps, equivariant geometry, and the interplay between symmetry and analysis on manifolds.
The structure of the 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 frameworks. We will then present our main results, detailing the derivation of the equations governing equivariant ( T )-harmonic maps and their implications. Finally, we will conclude with a discussion of potential future directions for research in this area.
The following references provide a comprehensive background for the topics discussed in this paper:
This extensive body of literature provides a solid foundation for the exploration of equivariant ( T )-harmonic maps and their geometric implications, setting the stage for the contributions presented in this paper.