Let $\gamma_i$, $i=1,2$ be two loops in $\mathbb R^3$. The Möbius cross energy of the pair is defined by $$ E(\gamma_1, \gamma_2)=\iint_{S^1\times S^1}\frac{|\gamma'_1(u)|\cdot|\gamma'_2(v)|}{|\gamma_1(u)-\gamma_2(v)|^2}du\,dv. $$ If I want to work in the round $S^3$ intrinsically, is there any formula for the Möbius cross energy?
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2$\begingroup$ Isn't your formula for $E(\gamma_1,\gamma_2)$ exactly what you are looking for? If not, why not? $\endgroup$– Ryan BudneyCommented May 26, 2021 at 3:08
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$\begingroup$ Just a warning. There have been a great deal of research on Mobius energy, O'Hara energy, Menger curvature etc. so unless you know exactly what you want to do, starting original research in this field might be difficult. $\endgroup$– Piotr HajlaszCommented May 26, 2021 at 3:17
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1$\begingroup$ @RyanBudney The `standard' Hopf link in the round $S^3$ is two great circles. Their distance is $\pi/2$ point-wisely. Therefore, if you use this intrinsic distance, the energy is not $2\pi^2$ (Which is the energy of the link in the Euclidean $R^3$). However, if you use the distance of $\mathbb R^4$ restricted on $S^3$, which is non-Riemannian, then you have $2\pi^2$. $\endgroup$– J. GECommented May 26, 2021 at 9:32
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$\begingroup$ I suspect you are using the word "distance" in a non-standard way. Perhaps you need to clarify what "distance" means, each time you state it. Moreover, what does "intrinsic" mean to you? I suspect your question has a fairly tautological answer once you notice that $|\gamma_1-\gamma_2|$ can be written in terms of the Hopf-Rinow metric on $S^3$. This involves a little trigonometry. $\endgroup$– Ryan BudneyCommented May 26, 2021 at 15:48
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1$\begingroup$ Not trivial, but I thought this material was standard in introductory differential geometry courses. The main observation is that $|\gamma_1-\gamma_2|$ is equal to $2 - 2\cos \theta$ where $\theta$ is the "intrinsic distance" or what is often known as the Hopf-Rinow metric distance between $\gamma_1$ and $\gamma_2$. This follows from the Hopf-Rinow theorem, knowledge of the geodesics in a sphere and the "law of cosines". $\endgroup$– Ryan BudneyCommented May 27, 2021 at 4:09
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