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 neighborhood Euclidean torus, and introduces the elemental innovations of Euler numbers for topological triangulations, and Riemannian metrics. The cautious dialogue of those classical examples presents scholars with an creation to the extra normal concept of curved areas built later within the ebook, as represented by way of embedded surfaces in Euclidean 3-space, and their generalization to summary surfaces outfitted 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 foremost issues to lifestyles and priceless examples and routines are integrated to help knowing. in the course of the emphasis is put on particular proofs, making this article perfect for any scholar with a simple historical past in research and algebra.
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Extra resources for Curved Spaces: From Classical Geometries to Elementary Differential Geometry
It is well known however that the space S 3 has no nontrivial covers; this property is equivalent to the property that S 3 is simply connected, namely that any closed path on S 3 may be continuously shrunk down to a point. Corresponding to the ﬁnite subgroups of SO(3), namely cyclic, dihedral, and the rotation groups of the tetrahedron, cube and dodecahedron, there are ﬁnite subgroups of SU (2) of twice the order. 19. The subgroup of SU (2) corresponding to a dihedral group D2n in SO(3) is called the dicyclic group; it contains a subgroup C2n , but the order 2 subgroup of D2n generated by the rotation of S 2 through an angle π about the x-axis lifts to a cyclic subgroup of order 4.
The above proof is again more widely applicable than just to the case of simple closed polygonal curves. Suppose, for instance, that γ is made up of circular arcs and line segments. As before, we choose a point P2 ∈ C with d (0, P2 ) maximum, which by the same argument as above may be assumed unique. If P2 is not a vertex, it must be an interior point of a circular arc between vertices P1 and P3 ; in the case where it is a vertex, we let P1 , P3 denote the vertices immediately before (respectively, after) P2 .
E. P and R antipodal. In that case, the line PQ on S also passes through R and so Proof d (P, R) = d (P, Q) + d (Q, R). In contrast to the Euclidean case, there is a rather useful second cosine formula, which may be used for instance to deduce the lengths of the sides of a spherical triangle from knowledge of its angles. 8 (Second cosine formula) With the notation as before, sin α sin β cos c = cos γ − cos α cos β. Proof With the conventions as before concerning the spherical triangle ABC, we denote by A , B and C the unit normals −n1 , −n2 , −n3 in the directions of, respectively, B × C, C × A and A × B — these are the inward pointing normals to the solid OABC.