Differential Geometry and Binary Operations by Nikita E. Barabanov, Abraham A. Ungar

“…We derive a large set of binary operations that are algebraically isomorphic to the binary operation of the Beltrami–Klein ball model of hyperbolic geometry, known as the Einstein addition. We prove that each of these operations gives rise to a gyrocommutative gyrogroup isomorphic to Einstein gyrogroup, and satisfies a number of nice properties of the Einstein addition. …”
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On the Universal Group PSL(2, C) by Benjamin Fine, Gerhard Rosenberger

“…The group Γ = PSL(2, C) arises in a wide variety of contexts; hyperbolic geometry, automorphic function theory, number theory and group theory. …”
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Isomorphism of Binary Operations in Differential Geometry by Nikita E. Barabanov

“…We consider smooth binary operations invariant with respect to unitary transformations that generalize the operations of the Beltrami–Klein and Beltrami–Poincare ball models of hyperbolic geometry, known as Einstein addition and Möbius addition. …”
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Machine learning holographic mapping by neural network renormalization group by Hong-Ye Hu, Shuo-Hui Li, Lei Wang, Yi-Zhuang You

“…We apply this approach to the complex ϕ^{4} theory in two-dimensional Euclidian space-time in its critical phase, and show that the emergent bulk geometry matches the three-dimensional hyperbolic geometry when geometric fluctuation is neglected.…”
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Hyperbolic matrix factorization improves prediction of drug-target associations by Aleksandar Poleksic

“…We see this as additional evidence that the hyperbolic geometry underpins large biological networks.…”
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Bootstrapping closed string field theory by Fırat, Atakan H.

“…Abstract The determination of the string vertices of closed string field theory is shown to be a conformal field theory problem solvable by combining insights from Liouville theory, hyperbolic geometry, and conformal bootstrap. We first demonstrate how Strebel differentials arise from hyperbolic string vertices by performing a WKB approximation to the associated Fuchsian equation, which we subsequently use it to derive a Polyakov-like conjecture for Strebel differentials. …”
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A study of complex network using hyperbolic triangle group by Taha, M. H. M., Zainuddin, Hishamuddin, Mohd Shah, Nurisya, Chan, Kar Tim

“…Recent research has hinted that networks that have an underlying hyperbolic geometry are able to capture these properties. …”
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Perturbative classical conformal blocks as Steiner trees on the hyperbolic disk by Konstantin Alkalaev, Mikhail Pavlov

“…Abstract We consider the Steiner tree problem in hyperbolic geometry in the context of the AdS/CFT duality between large-c conformal blocks on the boundary and particle motions in the bulk. …”
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The Quantum Nature of Color Perception: Uncertainty Relations for Chromatic Opposition by Michel Berthier, Edoardo Provenzi

“…This strategy permitted to reveal the importance of hyperbolic geometry in colorimetry. Starting from these premises, we show how Resnikoff’s construction can be extended to a geometrically rich quantum framework, where the concepts of achromatic color, hue and saturation can be rigorously defined. …”
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THE LOGARITHMIC SPIRAL AND ITS SPHERICAL COUNTERPART by Hellmuth Stachel, Giorgio Figliolini, Jorge Angeles

“…On the other hand, spherical loxodromes can also be seen as helical curves in the projective model of hyperbolic geometry, where the sphere serves as a Clifford surface. …”
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THE LOGARITHMIC SPIRAL AND ITS SPHERICAL COUNTERPART by Hellmuth STACHEL, Giorgio FIGLIOLINI, Jorge ANGELES

“…On the other hand, spherical loxodromes can also be seen as helical curves in the projective model of hyperbolic geometry, where the sphere serves as a Clifford surface. …”
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Maximal supersymmetry and B-mode targets by Renata Kallosh, Andrei Linde, Timm Wrase, Yusuke Yamada

“…We relate the issue of stability of inflationary trajectories in these models to tessellations of a hyperbolic geometry.…”
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THE LOGARITHMIC SPIRAL AND ITS SPHERICAL COUNTERPART by Hellmuth Stachel, Giorgio Figliolini, Jorge Angeles

“…On the other hand, spherical loxodromes can also be seen as helical curves in the projective model of hyperbolic geometry, where the sphere serves as a Clifford surface. …”
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Hyperbolic planforms in relation to visual edges and textures perception. by Pascal Chossat, Olivier Faugeras

“…We then extend the classical ring model to this case and show that its natural framework is the non-Euclidean hyperbolic geometry. This brings in the beautiful structure of its group of isometries and certain of its subgroups which have a direct interpretation in terms of the organization of the neural populations that are assumed to encode the structure tensor. …”
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No ensemble averaging below the black hole threshold by Jean-Marc Schlenker, Edward Witten

“…To demonstrate this requires some novel results about the renormalized volume of a hyperbolic three-manifold, which we prove using modern methods in hyperbolic geometry. Why then do any observables show apparent ensemble averaging? …”
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Vitreous Carbon, Geometry and Topology: A Hollistic Approach by Patrice Mélinon

“…With this in mind, special attention is paid to the hyperbolic geometry giving birth to triply periodic minimal surfaces. …”
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Topologically protected optical signal processing using parity–time-symmetric oscillation quenching by Yu Sunkyu, Piao Xianji, Park Namkyoo

“…Although a representative example is the knotted geometry of wavefunctions in reciprocal space for quantum Hall family and topological insulators, topological states have also been defined for other physical quantities, such as topologically distinct Fermi surfaces and enhanced lattice degrees of freedom in hyperbolic geometry. Here, we investigate a different class of topological states – topological geometry of dynamical state trajectories – in non-Hermitian and nonlinear optical dynamics, revealing topologically protected oscillation quenching mechanisms determined by parity–time (PT) symmetry. …”
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From the Lorentz Transformation Group in Pseudo-Euclidean Spaces to Bi-gyrogroups by Abraham A. Ungar

“…The gyrogroup and gyrovector space structure ‎of the ball Rcn‎‎, ‎in turn‎, ‎form the algebraic setting for the Beltrami-Klein ball model ‎of hyperbolic geometry‎, ‎which underlies the ball Rcn‎‎. ‎…”
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Practical service life assessment and rehabilitation strategy development for hyperbolic shell natural draft cooling towers by Gries Matthew B., Michols Kevin A., Lawler John S., Wagner Elizabeth I., Dennis Anthea, Kgatle Koketso, Pretorius Marita

“…Condition assessment of natural draft cooling towers is challenging due to their massive size, hyperbolic geometry, and plant operational considerations (i.e. schedule, risk and cost). …”
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DyLFG: A Dynamic Network Learning Framework Based on Geometry by Wei Wu, Xuemeng Zhai

“…In order to extract the structural information of network snapshots, we designed a new hyperbolic geometry processing layer, which is different from the previous literature. …”
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