“…The concept of harmonic maps with potential, was initially suggested by Ratto in [3] and recently developed by several authors : V. Branding [14], Jiang [12] and others. The notion of biharmonic maps with potential was studied by A. Mohammed Cherif and M. Djaa in 2017 [2], and by A. Zagane and S. Ouakass [4] in 2018.…”
In this note, we characterize the f -harmonic maps and bi-f -harmonic maps with potential. We prove that every bi-f -harmonic map with potential from complete Riemannian manifold, satisfying some conditions is a f -harmonic map with potential. More, we study the case of conformal maps between equidimensional manifolds.
“…The concept of harmonic maps with potential, was initially suggested by Ratto in [3] and recently developed by several authors : V. Branding [14], Jiang [12] and others. The notion of biharmonic maps with potential was studied by A. Mohammed Cherif and M. Djaa in 2017 [2], and by A. Zagane and S. Ouakass [4] in 2018.…”
In this note, we characterize the f -harmonic maps and bi-f -harmonic maps with potential. We prove that every bi-f -harmonic map with potential from complete Riemannian manifold, satisfying some conditions is a f -harmonic map with potential. More, we study the case of conformal maps between equidimensional manifolds.
“…A similar concept of a HM with potential was also introduced independently by Fardoun and Ratoo lately 21 . The development of the HM with potential has been conducted by many others 25 , 29 , 38 – 40 . Figure 1 The schematic of the extended harmonic map (EHM) theory.…”
One potential pathway to find an ultimate rule governing our universe is to hunt for a connection among the fundamental equations in physics. Recently, Ren et al. reported that the harmonic maps with potential introduced by Duan, named extended harmonic mapping (EHM), connect the equations of general relativity, chaos and quantum mechanics via a universal geodesic equation. The equation, expressed as Euler–Lagrange equations on the Riemannian manifold, was obtained from the principle of least action. Here, we further demonstrate that more than ten fundamental equations, including that of classical mechanics, fluid physics, statistical physics, astrophysics, quantum physics and general relativity, can be connected by the same universal geodesic equation. The connection sketches a family tree of the physics equations, and their intrinsic connections reflect an alternative ultimate rule of our universe, i.e., the principle of least action on a Finsler manifold.
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