Abstract
Sub-Riemannian spaces are spaces whose metric structure may be viewed as a constrained geometry, where motion is only possible along a given set of directions, changing from point to point. The simplest example of such spaces is given by the so-called Heisenberg group. The characteristic constrained motion of sub-Riemannian spaces has numerous applications in robotic control in engineering and neurobiology where it arises naturally in the functional magnetic resonance imaging (FMRI). It also arises naturally in other branches of pure mathematics as Cauchy Riemann geometry, complex hyperbolic spaces, and jet spaces. In this paper, we review the use of the relationship between Heisenberg geometry and Cauchy Riemann (CR) geometry. More precisely, we focus on the problem of the prescription of the scalar curvature using techniques related to the theory of critical points at infinity. These techniques were first introduced by Bahri, Bahri and Brezis for the Yamabe conjecture in the Riemannian settings.
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Acknowledgements
The author would like to express posthumously her deep gratitude to her Professor, the great mathematician Abbas Bahri, without whom this work would not have been accomplished. The valuable research supervision of Professor Bahri, during the planning and development of the resolution of the CR Yamabe Conjecture and his willingness to give his knowledge and time so generously has been very much appreciated. The author also would like to thank the staff of Rutgers University, New Jersey, USA, where most of this work was realized.
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Dedicated to the Memory of Professor Abbas Bahri.
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Gamara, N. Methods of the theory of critical points at infinity on Cauchy Riemann manifolds. Arab. J. Math. 6, 153–199 (2017). https://doi.org/10.1007/s40065-017-0180-6
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DOI: https://doi.org/10.1007/s40065-017-0180-6