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In the setting of a pseudo-Riemannian conformal manifold $(M^n, [g])$ where $n \geq 2$, one can define a distinguished family of curves called conformal circles, or conformal geodesics. These curves are characterized by a third-order differential equation in the case of non-null conformal circles. One classical problem is to determine whether conformal circles play a fundamental role in a conformal manifold by examining if a diffeomorphism is a conformal diffeomorphism, provided it preserves unparametrized conformal circles. I will outline the definition of a conformal circle and discuss its holographic interpretation, where the term "holography" originates from the physics community. In this talk, "holography" refers to geometric relations between a Poincaré-Einstein manifold and its conformal infinity. Additionally, I will share my results on both the classical problem and its holographic interpretation.  

 

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