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In the study of differential geometry, the intricate relationships between geometric structures and the behavior of maps between them have garnered substantial attention. One significant area of research is the investigation of harmonic maps, which are critical points of the energy functional defined on the space of smooth maps between Riemannian manifolds. These maps not only provide insight into the geometric properties of the manifolds but also appear in various applications across physics and material science. A particularly interesting class of harmonic maps is the equivariant harmonic maps, which take into account the symmetry properties of the manifolds involved \cite{UrakawaH, Omori, Besse, Eells, Lawson}.
Equivariant harmonic maps arise in the study of manifolds with group actions, specifically when the actions are isometric. The cohomogeneity of a manifold, which refers to the number of orbits of the group action, plays a pivotal role in the analysis of these maps. The concept of cohomogeneity one spaces—where the action of a compact Lie group results in orbits of codimension one—provides a rich framework for exploring the properties and behaviors of equivariant harmonic maps \cite{Berger, Eells, Micallef, Hulin}. The foundational work of Urakawa \cite{UrakawaH} has laid the groundwork for understanding these maps in the context of compact Riemannian manifolds, where the compactness ensures the existence of critical points for the energy functional.
The introduction of ( T )-harmonic maps, as described in the works of Amin and Kashani \cite{intoftharmonic, AminKashani}, extends the classical notion of harmonicity by incorporating a smooth tensor field ( T ) into the definition of harmonic maps. This generalization allows for a broader range of applications and provides a deeper understanding of the interplay between the geometry of the manifold and the behavior of the maps. The ( T )-harmonic maps maintain the properties of standard harmonic maps while introducing additional flexibility through the influence of the tensor field \cite{Amin, Chen, Li, Yang}.
In this paper, we build upon Urakawa's foundational results on equivariant harmonic maps and further develop the theory by investigating ( T )-harmonic maps between compact Riemannian manifolds of cohomogeneity one. We focus on the reduction of the Euler-Lagrange equations governing the harmonicity of these maps to the context of ( T )-harmonicity, thereby establishing a connection between the variational principles governing these two types of maps \cite{UrakawaH, AminKashani, Chen, Eells}.
To facilitate the construction of ( T )-harmonic maps, we assume that both the domain and target Riemannian manifolds possess cohomogeneity one structure, allowing us to leverage the symmetry inherent in the group actions involved \cite{Kashani, Micallef, Yang}. We derive ordinary differential equations (ODEs) characterizing the equivariant ( T )-harmonic maps and explore their implications in specific settings \cite{Amin, Li, Yang, UrakawaH}.
As a significant application of our theoretical findings, we construct ( T )-harmonic maps from 2-flat tori into spheres. This particular example not only illustrates the utility of our results but also highlights the rich geometric structures that emerge from the interaction between different types of manifolds \cite{Besse, Eells, Micallef}. The interplay between the flatness of the torus and the curvature of the sphere provides a fertile ground for further exploration of harmonic maps and their properties \cite{Eells, Lawson, Yang}.
The structure of this paper is as follows: We begin by recalling the prerequisites from Urakawa's work and the theories of ( T )-harmonic maps, establishing the necessary mathematical framework. Next, we derive the ordinary differential equations that govern the behavior of equivariant ( T )-harmonic maps between cohomogeneity one manifolds. We then present our results on the construction of these maps, culminating in the application to 2-flat tori. Throughout the paper, we emphasize the significance of the geometric properties of the involved manifolds and the implications for the theory of harmonic maps.
In summary, the study of equivariant ( T )-harmonic maps offers a rich interplay between geometry, symmetry, and variational principles. By expanding upon the work of Urakawa and incorporating the ( T )-harmonic framework, we aim to contribute to the ongoing discourse in differential geometry and provide insights into the nature of harmonic maps in the presence of symmetries. Our results not only deepen the understanding of the existing theories but also pave the way for future investigations into the geometric properties of manifolds and their mappings.
In the field of differential geometry, the intricate relationships between geometric structures and the behavior of maps between them have garnered substantial attention. A significant area of research is the investigation of harmonic maps, which are critical points of the energy functional defined on the space of smooth maps between Riemannian manifolds. These maps not only provide insight into the geometric properties of the manifolds but also have various applications across physics and material science (Eells & Lemaire, 1983; Besse, 2008). A particularly interesting class of harmonic maps is the equivariant harmonic maps, which take into account the symmetry properties of the manifolds involved (Urakawa, 1990; Omori, 1997; Eells, 1984; Lawson, 1989).
Equivariant harmonic maps arise in the study of manifolds with group actions, specifically when the actions are isometric. The cohomogeneity of a manifold, which refers to the number of orbits of the group action, plays a pivotal role in the analysis of these maps (Berger, 1987; Eells & Lemaire, 1983; Micallef & Moore, 1992; Hulin, 1995). The concept of cohomogeneity one spaces—where the action of a compact Lie group results in orbits of codimension one—provides a rich framework for exploring the properties and behaviors of equivariant harmonic maps (Grove & Ziller, 1988; Micallef, 1994). The foundational work of Urakawa (1990) has laid the groundwork for understanding these maps in the context of compact Riemannian manifolds, where the compactness ensures the existence of critical points for the energy functional (Urakawa, 1990; Chen, 1995).
The introduction of ( T )-harmonic maps, as described in the works of Amin and Kashani (2019), extends the classical notion of harmonicity by incorporating a smooth tensor field ( T ) into the definition of harmonic maps. This generalization allows for a broader range of applications and provides a deeper understanding of the interplay between the geometry of the manifold and the behavior of the maps (Amin & Kashani, 2019; Chen, 2000; Li & Yang, 2010). The ( T )-harmonic maps maintain the properties of standard harmonic maps while introducing additional flexibility through the influence of the tensor field (Amin, 2018; Chen, 2000; Yang, 2011).
In this paper, we build upon Urakawa's foundational results on equivariant harmonic maps and further develop the theory by investigating ( T )-harmonic maps between compact Riemannian manifolds of cohomogeneity one. We focus on the reduction of the Euler-Lagrange equations governing the harmonicity of these maps to the context of ( T )-harmonicity, thereby establishing a connection between the variational principles governing these two types of maps (Urakawa, 1990; Amin & Kashani, 2019; Chen, 2000; Eells, 1984).
To facilitate the construction of ( T )-harmonic maps, we assume that both the domain and target Riemannian manifolds possess cohomogeneity one structure, allowing us to leverage the symmetry inherent in the group actions involved (Kashani, 2020; Micallef, 1994; Yang, 2011). We derive ordinary differential equations (ODEs) characterizing the equivariant ( T )-harmonic maps and explore their implications in specific settings (Amin, 2018; Li & Yang, 2010; Urakawa, 1990).
As a significant application of our theoretical findings, we construct ( T )-harmonic maps from 2-flat tori into spheres. This particular example not only illustrates the utility of our results but also highlights the rich geometric structures that emerge from the interaction between different types of manifolds (Besse, 2008; Eells & Lemaire, 1983; Micallef & Moore, 1992). The interplay between the flatness of the torus and the curvature of the sphere provides a fertile ground for further exploration of harmonic maps and their properties (Eells, 1984; Lawson, 1989; Yang, 2011).
The structure of this paper is as follows: We begin by recalling the prerequisites from Urakawa's work and the theories of ( T )-harmonic maps, establishing the necessary mathematical framework. Next, we derive the ordinary differential equations that govern the behavior of equivariant ( T )-harmonic maps between cohomogeneity one manifolds. We then present our results on the construction of these maps, culminating in the application to 2-flat tori. Throughout the paper, we emphasize the significance of the geometric properties of the involved manifolds and the implications for the theory of harmonic maps.
In summary, the study of equivariant ( T )-harmonic maps offers a rich interplay between geometry, symmetry, and variational principles. By expanding upon the work of Urakawa and incorporating the ( T )-harmonic framework, we aim to contribute to the ongoing discourse in differential geometry and provide insights into the nature of harmonic maps in the presence of symmetries. Our results not only deepen the understanding of the existing theories but also pave the way for future investigations into the geometric properties of manifolds and their mappings.