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.
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 ...
I also know that the hyperbolic functions parametrize the unit hyperbola $ (\cosh a, \sinh a)$ and that $\frac {a} {2}$ corresponds to the area of the region bounded by the hyperbola, the rays from the origin, and the point $ (\cosh a, \sinh a)$.
Working in the hyperbolic plane, let an $n$-gon be a finite sequence of points $ [a_0,...,a_n=a_0]$. Say that two $n$ -gons $ [...a_i...], [...b_i...]$ are quasisimilar iff
Why are the Partial Differential Equations so named? i.e, elliptical, hyperbolic, and parabolic. I do know the condition at which a general second order partial differential equation becomes these,...