By P. M. H. Wilson
ISBN-10: 0793164672
ISBN-13: 9780793164677
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 techniques of Euler numbers for topological triangulations, and Riemannian metrics. The cautious dialogue of those classical examples offers scholars with an creation to the extra basic conception of curved areas built later within the publication, as represented by means 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. various diagrams support carry the main issues to lifestyles and worthy examples and workouts are integrated to assist knowing. during the emphasis is put on specific proofs, making this article perfect for any scholar with a uncomplicated heritage in research and algebra.
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Extra resources for Curved Spaces - From Classical Geometry to Elementary Differential Geometry
Sample text
Through any two non-antipodal points P, Q on S, there exists precisely one line (namely, we intersect S with the plane determined by OPQ). 1 25 26 SPHERIC AL GEOMETRY P and Q are antipodal). In this chapter, we shall always use d to denote this distance function on the sphere. − → −→ Note that d (P, Q) is just the angle between P = OP and Q = OQ, and hence is just cos−1 (P, Q), where (P, Q) = P · Q is the Euclidean inner-product on R 3 . For reasons which will become clear later, the spherical lines are also sometimes called the geodesics or geodesic lines on S 2 .
How many of the isometries is this group are rotated reflections (and not pure reflections)? Describe these rotated reflections geometrically, by identifying the axes of rotation and the angles of rotation. Given F a closed subset of a metric space (X , d ), show that the real-valued function d (x, F) := inf {d (x, y) : y ∈ F} is continuous, and strictly positive on the complement of F. 17 is strictly positive. 4 via lengths of curves. For any curve γ : [a, b] → X , we denote by ld0 (γ ), respectively ld (γ ), the lengths of γ as defined with respect to the two metrics.
Proof The property of the hemisphere we use is that of it being convex, in the sense defined above. We prove the formula by induction on n, the case n = 3 following from Gauss–Bonnet. We show that there is always an internal diagonal (that is, a spherical line segment joining non-adjacent vertices whose interior is contained in the interior Int of ); this diagonal then divides into two polygons, both with strictly less than n sides, and the result follows by induction. We assume without loss of generality that is in the southern hemisphere.
Curved Spaces - From Classical Geometry to Elementary Differential Geometry by P. M. H. Wilson
by Paul
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