By P. M. H. Wilson

This self-contained textbook provides an exposition of the well known classical two-dimensional geometries, reminiscent of Euclidean, round, hyperbolic, and the in the community Euclidean torus, and introduces the elemental suggestions of Euler numbers for topological triangulations, and Riemannian metrics. The cautious dialogue of those classical examples presents scholars with an advent to the extra basic concept of curved areas built later within the e-book, as represented by way of embedded surfaces in Euclidean 3-space, and their generalization to summary surfaces outfitted with Riemannian metrics. topics operating all through comprise these of geodesic curves, polygonal approximations to triangulations, Gaussian curvature, and the hyperlink to topology supplied by way of the Gauss-Bonnet theorem. a number of diagrams support convey the main issues to existence and worthy examples and workouts are integrated to help figuring out. through the emphasis is put on particular proofs, making this article excellent for any scholar with a simple historical past in research and algebra.

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Circles on S2 Given an arbitrary point P on S 2 , and 0 ≤ ρ < π, we may consider the locus of points on S 2 whose spherical distance from P is ρ. This is what we mean by a circle 46 SPHERIC AL GEOMETRY in spherical geometry. We may always rotate the sphere so that the point P is at the north pole, as shown in the ﬁgure below. r sin r Hence it is clear that the circle is also a Euclidean circle, of radius sin ρ, and that it is the intersection of a plane with S 2 . Conversely, any plane whose intersection with S 2 consists of more than one point, cuts out a circle.

In 32 SPHERIC AL GEOMETRY summary therefore, we have seen that Isom(S 2 ) is naturally identiﬁed with the group O(3, R). The results we proved about O(3, R) in Chapter 1 will therefore have precise counterparts for Isom(S 2 ). We deﬁne a reﬂection of S 2 in a spherical line l (a great circle, say l = H ∩ S 2 for some plane H passing through the origin) to be the restriction to S 2 of the isometry RH of R 3 , the reﬂection of R 3 in the hyperplane H . It therefore follows immediately from results in the Euclidean case that any element of Isom(S 2 ) is the composite of at most three such reﬂections.

The icosahedron (with 12 vertices and 20 triangular faces) is dual to the dodecahedron, and so has the same rotation and full symmetry groups. A straightforward, albeit slightly long, argument using elementary group theory [4], shows that we have now accounted for all the ﬁnite subgroups of SO(3). The ﬁnite subgroups of SO(3) are of isomorphism types Cn for n ≥ 1, D2n for n ≥ 2, A4 , S4 , A5 , the last three being the rotation groups arising from the regular solids. 12 We comment now that −I ∈ O(3) \ SO(3), and so if G is a ﬁnite subgroup of SO(3), then H = C2 × G is a subgroup of O(3) of twice the order, with elements ±A for A ∈ G.