By P. M. H. Wilson
This self-contained textbook provides an exposition of the well known classical two-dimensional geometries, resembling Euclidean, round, hyperbolic, and the in the community Euclidean torus, and introduces the fundamental innovations of Euler numbers for topological triangulations, and Riemannian metrics. The cautious dialogue of those classical examples offers scholars with an advent to the extra basic concept of curved areas constructed later within the e-book, as represented by way of embedded surfaces in Euclidean 3-space, and their generalization to summary surfaces built with Riemannian metrics. topics operating all through comprise these of geodesic curves, polygonal approximations to triangulations, Gaussian curvature, and the hyperlink to topology supplied through the Gauss-Bonnet theorem. various diagrams support carry the foremost issues to existence and important examples and routines are integrated to assist realizing. through the emphasis is put on specific proofs, making this article excellent for any pupil with a easy historical past in research and algebra.
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Additional info for Curved Spaces: From Classical Geometries to Elementary Differential Geometry
18 EUCLIDEAN GEOMETRY Let γ : [a, b] → R 2 be a simple closed polygonal curve, with C ⊂ R 2 denoting the image γ ([a, b]). Then R 2 \ C has at most two path connected components. 17 Proof For each P ∈ C, we can ﬁnd an open ball B = B(P, ε) such that C ∩ B consists of two radial linear segments (often a diameter). 15, and is contained in the union of such open balls. 14, we deduce that C is contained in the union U of some ﬁnite subcollection of these balls, say U = B1 ∪ · · · ∪ BN . Our assumptions imply that U \ C consists of two path connected components U1 and U2 ; if one travels along the curve γ , then one of these components will always be to the left, and the other always to the right.
The area of is π/6. is π/n. 34 SPHERIC AL GEOMETRY • • ( p, q, r) = (2, 3, 4). The area of ( p, q, r) = (2, 3, 5). The area of is π/12. is π/30. The fact that S 2 (of area 4π ) is tessellated by the images of under G then implies that G has order 4n, 24, 48 and 120 in these cases. It is then straightforward to check that in the ﬁrst case G is C2 × D2n , and in the remaining cases it is the full symmetry group of the tetrahedron, cube and dodecahedron, respectively. The tessellation of S 2 arising in the ﬁrst case is clear.
Prove that the formula for the area of a spherical polygon remains true for polygons not necessarily contained in an open hemisphere. [Hint: A limiting argument veriﬁes the formula for spherical polygons contained in a closed hemisphere. 1 Geometry of the torus Before starting on the main topics of this chapter, we introduce another geometry, that of the locally Euclidean torus. The torus T = T 2 may be deﬁned most easily as follows. As a set it is just whose points are represented by (x, y) ∈ R 2 modulo the equivalence relation (x1 , y1 ) ∼ (x2 , y2 ) ⇐⇒ x2 − x1 ∈ Z and y2 − y1 ∈ Z.
Curved Spaces: From Classical Geometries to Elementary Differential Geometry by P. M. H. Wilson