By P. M. H. Wilson
This self-contained 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 elemental strategies of Euler numbers for topological triangulations, and Riemannian metrics. The cautious dialogue of those classical examples offers scholars with an creation to the extra common conception 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 built with Riemannian metrics. topics working 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. a variety of diagrams support carry the foremost issues to lifestyles and precious examples and workouts are incorporated to help figuring out. through the emphasis is put on specific proofs, making this article perfect for any pupil with a uncomplicated history in research and algebra.
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Additional resources for Curved Spaces: From Classical Geometries to Elementary Differential Geometry
4 (Spherical Pythagoras theorem) When γ = π 2, cos c = cos a cos b. There is also a formula corresponding to the Euclidean sine formula. 5 (Spherical sine formula) With the notation as above, sin b sin c sin a = = . sin α sin β sin γ 28 SPHERIC AL GEOMETRY Proof We use the vector identity (A × C) × (C × B) = (C · (B × A))C. In our case, the left-hand side of this equation is −(n1 ×n2 ) sin a sin b. Clearly n1 ×n2 is a multiple of C, and one veriﬁes easily that n1 × n2 = C sin γ . Therefore, equating the multiples of C, we deduce that C · (A × B) = sin a sin b sin γ .
This is only a minor change, since only one side 36 SPHERIC AL GEOMETRY could have length ≥ π (otherwise adjacent sides would meet twice, and we would not have a triangle). If however one of the sides has length ≥ π , we can subdivide the triangle into two smaller ones, whose sides have length less than π . Applying Gauss–Bonnet to the two smaller triangles and adding, the area of the original triangle is still α + β + γ + π − 2π = α + β + γ − π . We now extend the Gauss–Bonnet to spherical polygons on 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. We recall in passing that an exactly analogous fact held for the isometries of the Euclidean plane R 2 . There is moreover an index two subgroup of Isom(S 2 ) corresponding to the subgroup SO(3) ⊂ O(3); these isometries are just the rotations of S 2 , and are the composite of two reﬂections.
Curved Spaces: From Classical Geometries to Elementary Differential Geometry by P. M. H. Wilson