Hyperbolic Geometry (2nd Edition) (Springer Undergraduate by James W. Anderson

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.

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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.

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