There is some general meaning of angle in geometry?

Question

The other day I discovered the concept of hyperbolic angle to denote angles in hyperbolic geometry, as the half of the area between the hyperbola defined by $x^2-y^2=1$ and the $x$-axis.

To be honest I dont know very much about the general concept of geometry in mathematics, so I cant grasp the meaning of the above, if it have some intuitive meaning more than an analogy to the angle and the circle $x^2+y^2=1$.

Where I can understand something is in the realm of analysis or linear algebra.

My questions:

1. There is a general concept of angle in mathematics? Not only applicable to euclidean-geometries if not to any other kind of geometries.

2. If so, it can be defined in analytical terms?

3. There is a good reference about this topic understandable to someone with (some) background on analysis or linear algebra?

Answer 1

There are two answers to your question.

1. Euclidean and hyperbolic geometry are instances of Riemannian geometry. I am not sure you have enough background to understand what this means but if you have taken vector calculus, you know about (regular) surfaces in 3-d space. At every point $p$ on such a surface $S$ there is a tangent plane $T_pS$ which is a certain 2-dimensional linear subspace passing through $p$. Vectors in $T_pS$ are called tangent vectors to $S$ at $p$. Given two nonzero vectors $u, v\in T_pS$ one defines the angle between $u, v$ as the ordinary Euclidean vector. One also has the notion of the dot product $u\cdot v$.

Riemannian geometry generalizes this picture to "higher dimensions" with surfaces replaced by higher-dimensional objects called manifolds. A Riemannian metric on a manifold $M$ is a choice of an inner product ("dot product") for vectors $u, v\in T_pM$. Given this, one defines the angle between (nonzero) vectors $u, v\in T_pM$ as $$ arccos\left(\frac{u\cdot v}{|u| ~|v|}\right) $$
Vectors $u, v$ as above appear as velocity vectors for curves $\alpha, \beta$ in $M$ starting at $p$. Hence, one defines the angle between such curves (at $p$).

My favorite reference is:

M. Do Carmo, Riemannian Geometry.

2. The concept of Riemannian metrics (manifolds) has further generalization, called metric spaces. Under some conditions on metric spaces, one can define angles between (geodesic) curves in such spaces. However, this definition is likely to be beyond your level of comfort.

See for instance

D.Burago, S.Ivanov, A Course in Metric Geometry.