Homework 1 | Due January 24 (Tuesday)
1
Read : Stahl, Chapters 1.1, 1.2, 1.3, 1.4, 2.1. 1. Let X be a geometry, and let f and g be isometries of X. (a) Prove that f is invertible and f -1 is an isometry of X. (b) Prove that f g is an isometry of X. For pr
MthSc 481: Topics in Geometry & Topology
Spring 2012
Martin Hall M-103, TTh 8:00-8:50am
Instructor
Matthew Macauley ([email protected]) Office: Martin Hall O325 Phone: (864) 6561838 Office Hours: T 3:004:00, Th 2:003:00, or by appointment A Gateway to M
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Homework 6 | Due March 15 (Thursday)
1
Read : Stahl, Chapters 9.1, 9.2. 1. Prove that the rotations R0, and R0,- are conjugate in Isom(E2 ). 2. Prove that the glide reflections AB and CD are conjugate in Isom(E2 ) if and only if d(A, B) = d(C, D). 3. Let
Homework 5 | Due March 6 (Tuesday)
1
Read : Stahl Chapters 6.1, 6.2, 7.1, 7.2, 8.1, 8.2, 8.3. 1. Prove that if 0 < < 2 then there is a hyperbolic quadrilateral whose angles sum to . 2. For a, b > 0, let R(a, b) be the Euclidean rectangle with corners (0,
Homework 4 | Due February 21 (Tuesday)
1
Read : Stahl, Chapters 4.4., 5.1, 5.2, 5.3, 5.4. 1. Find the Euclidean center and radius of the circle that has hyperbolic center (5, 4) and radius 3. 2. Prove that the hyperbolic circumference of a circle with hyp
Homework 3 | Due February 7 (Tuesday)
1
Read : Stahl, Chapters 4.1, 4.2, 4.3. 1. In this problem, we will prove Proposition 3.1.3 in Stahl in a more rigorous and elegant fashion, using only the analytic methods we developed in class. (a) Let f be a Euclid
Homework 2 | Due January 31 (Tuesday)
1
Read : Stahl, Chapters 2.2, 2.3, 2.4, 2.5, 3.1. 1. In this problem, use only the analytic definition of Euclidean geometry. Let v be a unit vector in R2 , and define rv (x) = x - 2(x v)v. (a) Prove (analytically) th
Homework 9 | Due April 17 (Tuesday)
1
1. Let f, g Isom(H2 ) be orientation-preserving, and suppose that f is parabolic. Prove
that either there exists x R such that f and g both x x, or f n g is an isometry of
hyperbolic type for some value of n. [Hint :