Mathematics 265: Differential Geometry Assignments

Assignments

  1. Let X be the set of points in the plane, and consider the (possible) distance function between 2 points (x1,y1) and (x2,y2) be given by d=max(|x1-x2|,|y1-y2|). Is (X,d) a metric space? If so, prove it; if not, prove that it is not.
  2. Let X be the set of points in the plane, and consider the (possible) distance function between 2 points (x1,y1) and (x2,y2) be given by d=1 unless x1=x2 and y1=y2, in which case d=0. Is (X,d) a metric space? If so, prove it; if not, prove that it is not.
  3. We gave the integers Z the p-adic metric d(m,n)=p^{-k} if the highest power of p that m-n is divisible by is k. Extend this metric to the rational numbers Q.
  4. Let d(x,y) be the "straight-line" distance between points on the unit sphere discussed in lecture in connection with the set of lines passing through the origin in R^3, and let d'(x,y) be the visual angle distance on the same set of lines, also discussed in class. Is there a function f:R->R such that f(d(x,y))=d'(x,y) for all x,y? If so, find it.
  5. For each of the following metric spaces, do geodesic curves exist between all points? For those that do, are they unique? What is the diameter of each?
    1. R^2, the plane, with the taxicab metric
    2. R^2, the plane, with the max of abs values metric ("box" metric from problem 1)
    3. R^2, with the discrete "0-1" metric from problem 2.
    4. S^2, the unit sphere in R^3 with the angle-measure metric
    5. S^2, with the "straight-line" metric
    6. Z^2, the set of integer lattice points in the plane with the "taxicab" metric
    7. Z^2 union ZxR union RxZ, the set of integer lattice lines in the plane with the "straight-line" metric
    8. Z^2 union ZxR union RxZ, the set of integer lattice lines in the plane with the "taxicab" metric
    9. Z^2, the set of integer lattice points in the plane with the "straight-line" metric
    10. A torus in R^3 with the intrinsic metric
    11. A torus in R^3 with the metric from R^3
    12. R^2 mod Z^2, points in the plane under the equivalence relation (x1,y1)=(x2,y2) if x1-x2 and y1-y2 are integers
    13. The unit cube in R^3, with the intrinsic metric
    14. The unit cube in R^3 with the straight line metric
    15. The set of lines in R^3 with the "angle" metric
    16. The set of lines in R^3 with the "straight-line" metric (the minimum straight-line distance between the pairs of interection points on the unit sphere)
    17. The set of lines in R^4 with the "angle" metric
    18. The set of lines in R^4 with the "straight-line" metric.
  6. Exercises #1, #2 in section 2.5 of Ramsey and Richtmeyer
  7. Find the paths which minimize length in the "split-plane" metric space, where d=normal Euclidean metric for y>=0 and d= 2x normal Euclidean metric for y<0.
  8. Find the equation of the geodesics given the coordinates of two points in the Poincare upper half-plane model of the hyperbolic plane.
  9. Find the intersection of the lines between two pairs of points in the upper half-plane model.
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