By P. M. H. Wilson

This self-contained 2007 textbook offers an exposition of the well known classical two-dimensional geometries, resembling Euclidean, round, hyperbolic, and the in the neighborhood Euclidean torus, and introduces the fundamental 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 common idea of curved areas constructed later within the booklet, as represented by way of embedded surfaces in Euclidean 3-space, and their generalization to summary surfaces outfitted with Riemannian metrics. subject matters working all through comprise these of geodesic curves, polygonal approximations to triangulations, Gaussian curvature, and the hyperlink to topology supplied by means of the Gauss-Bonnet theorem. a variety of diagrams support deliver the major issues to existence and invaluable examples and routines are incorporated to help figuring out. through the emphasis is put on particular proofs, making this article excellent for any scholar with a easy history in research and algebra.

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**Example text**

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. Recall that the great circles just correspond to the planes passing though the origin. 6, we calculate the area of such a circle to be 2π(1 − cos ρ) = 4π sin2 (ρ/2), which is always less than the area πρ 2 from the Euclidean case, and for small ρ may be expanded as πρ 2 1 − 1 2 ρ + O(ρ 4 ) .

Show that the Möbius transformation corresponding (under the stereographic projection map) to a rotation of S 2 through a non-zero angle has exactly two ﬁxed points z1 and z2 , where z2 = −1/¯z1 . e. for z = z2 , we have that T n z → z1 as n → ∞. For any ﬁnite set of points in the Euclidean plane, show that there is a unique circle of minimum radius which encloses them (some will of course lie on the circle). 16 49 solid, show that there is no such unique spherical circle which encloses them all.

The other two side lengths are π − β and π − γ . To ﬁnd the angles of the polar triangle, we observe that the polar of the polar triangle is our original triangle; it is clear for instance that the unit vector in the direction B × C must be ±A, and one then easily veriﬁes that it is A. The original triangle had side lengths a, b and c; thus the angles of the polar triangle have to be π −a, π −b and π −c. The second cosine formula is then deduced simply by applying the ﬁrst cosine formula to the polar triangle.