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 options of Euler numbers for topological triangulations, and Riemannian metrics. The cautious dialogue of those classical examples presents scholars with an advent to the extra normal conception of curved areas constructed later within the ebook, as represented by means of embedded surfaces in Euclidean 3-space, and their generalization to summary surfaces built with Riemannian metrics. issues operating all through contain these of geodesic curves, polygonal approximations to triangulations, Gaussian curvature, and the hyperlink to topology supplied by means of the Gauss-Bonnet theorem. quite a few diagrams support convey the major issues to lifestyles and important examples and routines are incorporated to help figuring out. during 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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Additional info for Curved Spaces: From Classical Geometries to Elementary Differential Geometry
In the case of D4 = C2 × C2 , the dicyclic group we obtain is just the well-known quaternion group of order eight. The subgroups of SU (2) of orders 24, 48 and 120, corresponding to the three rotation groups of regular solids, are usually called the binary tetrahedral, binary octahedral and binary icosahedral groups. 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.
Namely, π(P) is the point of intersection of the line through N and P with C, where C is identiﬁed as the plane z = 0, and where we deﬁne π(N ) := ∞. Clearly, π is a bijection. N = (0,0,1) P π(P ) Using the geometry of similar triangles, we can produce an explicit formula for π , namely π(x, y, z) = x + iy 1−z 40 SPHERIC AL GEOMETRY since in the diagram below r R = 1−z 1 , and so R = z r r 1−z . R What happens if we project instead from the south pole? 17 If π : S 2 → C∞ denotes the stereographic projection from the south pole, then π (P) = 1/π(P) for any P ∈ S 2 .
Let l, l be two distinct lines in R 2 , meeting at a point P with an angle α. Show that the composite of the corresponding reﬂections Rl Rl is a rotation about P through an angle 2α. If l, l are parallel lines, show that the composite is a translation. Give an example of an isometry of R 2 which cannot be expressed as the composite of less than three reﬂections. Let R(P, θ ) denote the clockwise rotation of R 2 through an angle θ about a point P. If A, B, C are the vertices, labelled clockwise, of a triangle in R 2 , prove that the composite R(A, θ )R(B, φ)R(C, ψ) is the identity if and only if θ = 2α, φ = 2β and ψ = 2γ , where α, β, γ denote the angles at, respectively, the vertices A, B, C of the triangle ABC.
Curved Spaces: From Classical Geometries to Elementary Differential Geometry by P. M. H. Wilson
Categories: Algebraic Geometry