By James W. Anderson

The geometry of the hyperbolic aircraft has been an energetic and interesting box of mathematical inquiry for many of the previous centuries. This ebook offers a self-contained advent to the topic, appropriate for 3rd or fourth 12 months undergraduates. the elemental process taken is to outline hyperbolic strains and strengthen a ordinary team of adjustments protecting hyperbolic traces, after which learn hyperbolic geometry as these amounts invariant less than this crew of transformations.

Topics lined comprise the higher half-plane version of the hyperbolic airplane, Möbius adjustments, the final Möbius workforce, and their subgroups keeping the higher half-plane, hyperbolic arc-length and distance as amounts invariant lower than those subgroups, the Poincaré disc version, convex subsets of the hyperbolic airplane, hyperbolic region, the Gauss-Bonnet formulation and its applications.

This up-to-date moment version additionally features:

an increased dialogue of planar types of the hyperbolic aircraft coming up from complicated analysis;

the hyperboloid version of the hyperbolic plane;

brief dialogue of generalizations to raised dimensions;

many new exercises.

The kind and point of the ebook, which assumes few mathematical must haves, make it an incredible advent to this topic and offers the reader with a company grab of the ideas and methods of this gorgeous a part of the mathematical panorama.

**Read Online or Download Hyperbolic Geometry (2nd Edition) (Springer Undergraduate Mathematics Series) PDF**

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**Extra resources for Hyperbolic Geometry (2nd Edition) (Springer Undergraduate Mathematics Series)**

**Example text**

Prove that p is not a homeomorphism of C. We close this section with a crude classification of M¨obius transformations, based on the number of fixed points. A fixed point of the M¨ obius transformation m is a point z of C satisfying m(z) = z. Suppose that m is not the identity. 30 Hyperbolic Geometry We saw earlier in this section that for m(z) = and so m(∞) = ∞ if and only if c = 0. az+b cz+d , we have that m(∞) = ac , If c = 0, then m(z) = ad z + db , and the fixed point of m in C is the solution to the equation m(z) = ad z + db = z.

Z2 − z4 ) (z3 − z1 ) We note here that all possible choices of cross ratio, such as the three described in this section, are all closely related. 16 Express the two cross ratios [z1 , z2 ; z3 , z4 ]2 and [z1 , z2 ; z3 , z4 ]3 in terms of the standard cross ratio [z1 , z2 ; z3 , z4 ]. 1, in terms of the number of fixed points, is as we wrote at the time crude and can be considerably refined. Before getting into the refinement of this classification, we introduce a notion of sameness for M¨obius transformations.

Of all the hyperbolic lines through p, there is exactly one that contains q in its boundary at infinity, namely, the hyperbolic line contained in the Euclidean line {z ∈ C | Re(z) = Re(p)}. The statement about uniqueness follows from the observation that no hyperbolic line contained in a Euclidean circle contains ∞ in its boundary at infinity. Suppose that q ̸= ∞ and that Re(p) = Re(q). Then, the hyperbolic line contained in the Euclidean line {z ∈ C | Re(z) = Re(p)} is the unique hyperbolic line through p that contains q in its boundary at infinity.