Complex hyperbolic geometry investigates spaces that combine the subtleties of complex structures with non‐Euclidean, negatively curved metrics, offering rich terrain for both theoretical exploration ...
Complex hyperbolic geometry studies spaces that combine the rich structure of complex manifolds with the intriguing features of hyperbolic curvature. At its heart lies the complex hyperbolic space, a ...
geometry - What is the relevance of hyperbolic sine and cosine? What is ...
triangles - A notion of similarity in hyperbolic geometry - Mathematics ...
Any manifold is the quotient of its universal cover by its fundamental group, so this statement is a special case of a general principle. So what you are looking for is the statement that a complete simply connected manifold of sectional curvature $-1$ is isometric to the hyperbolic space. This is a basic result in Riemannian geometry and can be found for instance in do Carmo, Manfredo ...
While $\mathbb H^n$ is not really an affine space, the general equation for hyperbolic hyperplanes is just a manifestation of this broad correspondence between affine spaces (inhomogeneous) and vector spaces with one more dimension (homogeneous), which also manifests itself in algebra and algebraic geometry as homogenization of polynomials.
Reducing redundant information to find simplifying patterns in data sets and complex networks is a scientific challenge in many knowledge fields. Moreover, detecting the dimensionality of the data is ...
Complex networks, ranging from biological systems to technological and social infrastructures, have long intrigued researchers with their intricate connectivity and emergent behaviours. Recent ...
Tessellations aren’t just eye-catching patterns—they can be used to crack complex mathematical problems. By repeatedly reflecting shapes to tile a surface, researchers uncovered a method that links ...