Differential Geometry

New PDF release: A First Course in Geometric Topology and Differential

By Ethan D. Bloch

ISBN-10: 0817638407

ISBN-13: 9780817638405

The individuality of this article in combining geometric topology and differential geometry lies in its unifying thread: the idea of a floor. With a variety of illustrations, workouts and examples, the scholar involves comprehend the connection among sleek axiomatic procedure and geometric instinct. The textual content is stored at a concrete point, 'motivational' in nature, keeping off abstractions. a couple of intuitively attractive definitions and theorems relating surfaces within the topological, polyhedral, and delicate circumstances are offered from the geometric view, and element set topology is particular to subsets of Euclidean areas. The remedy of differential geometry is classical, facing surfaces in R3 . the cloth here's available to math majors on the junior/senior point.

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Extra info for A First Course in Geometric Topology and Differential Geometry

Example text

A Schottky group of rank p is a subgroup of G which has a collection of non-elliptic, non-trivial generators {g1 , . . , gp } satisfying the following condition: there exists a point 0 in D such that the closures in D = D ∪ D(∞) of the sets D0 (gi±1 ) = D(gi±1 ), for i = 1, . . , p, satisfy (D(gi ) ∪ D(gi−1 )) ∩ (D(gj ) ∪ D(gj−1 )) = ∅, for all i = j in {1, . . , p}. Let S(g1 , . . , gp ) denote such a group whose collection of generators is {g1 , . . , gp }. In the rest of this discussion, in order to avoid notational clutter, we will restrict ourselves to the case where p = 2.

16. The first two surfaces are topologically Fig. 25. equivalent. Both are homeomorphic to a cylinder. From the metric point of view, however, there are some differences. To highlight these differences, choose a generator g of Γ . Define the curve c = [g −1 (i), i]h ∪ [i, g(i)]h . Let c be the intersection of c with the Dirichlet domain Di (Γ ). Now (referring to Fig. 25) remove c from Di (Γ ). In the first case, the curve c is the segment [2i, 1/2i]h , thus c splits the domain into two subdomains of infinite area.

Fig. 20. h(z) = 2z Fig. 21. , for any z and z in Hz0 (γ), the hyperbolic segment [z, z ]h is included in this set). It follows that the intersection of all Hz0 (γ) is also convex. Define Hz0 (γ). Dz0 (Γ ) = γ∈Γ γ=Id This set is called the Dirichlet domain of Γ centered at z0 . 11. A Dirichlet domain is a convex fundamental domain of Γ . 12. 11. ) Let us examine the boundary of the Dirichlet domain Dz0 (Γ ) in H. 13. The set Dz0 (Γ )− Dz0 (Γ ) is in the union of the perpendicular bisectors of the segments [z0 , γ(z0 )]h , with γ in Γ − {Id}.

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A First Course in Geometric Topology and Differential Geometry by Ethan D. Bloch


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