Download MATH 301 Survey of Geometries Homework Problems – Week 5

MATH 301 Survey of Geometries
Spring 2011
Homework Problems – Week 5
7.1 A lune, or biangle, is any of the four regions on a sphere bounded by two distinct great circles.
(a) The two sides of each interior angle of a triangle ∆ on a sphere determine two congruent
lunes with lune angle the same as the interior angle. Show how the three pairs of lunes
determined by the three interior angles α, β, γ cover the sphere, with some overlap. (See
Figure 7.2, p. 90.)
(b) Find a formula for the area of a lune with lune angle θ in terms of θ and the (surface)
are of the sphere (of radius ρ), which you can call Sρ . Use radian measure for the angles.
(Hint: first try the special angles θ = π or π2 .)
(c) Find a formula for the area of a triangle on a sphere of radius ρ.
7.2 Next we investigate a formula for the area of a triangle on a hyperbolic plane.
(a) Consider as large a triangle as possible on a hyperbolic plane. Describe some of the
features of such a triangle. (Consult your classmates about what they find and compare
notes!)
(b) A 2/3-ideal triangle is a figure on a hyperbolic plane consisting of an angle formed by
two rays with a common endpoint (the vertex of the 2/3-ideal triangle) and a line simultaneously asymptotic to each of the two sides of the given angle. (See Figures 7.3 and 7.4
on pp. 92-93.) Show that (on the same hyperbolic plane) all 2/3-ideal triangles with the
same angle θ are congruent to each other.
(c) The radius ρ of a hyperbolic plane is the radius (in the sense of Figure 5.9, p. 69) of the
annular curves that run perpendicular to the radial lines in the plane (or the inner radius
of the strips used to piece together a model of the plane, as in Figures 5.2-5.7, pp. 62-65).
Where Aρ (θ) denotes the area of a 2/3-ideal triangles with angle θ, show that this area is
an additive function of the angle, meaning that Aρ (α + β) = Aρ (α) + Aρ (β). See Figure
7.5, p. 94, for a hint.
(d) Following the discussion on pp. 94-95 as a guide, (i) argue that all ideal triangles (on
the same hyperbolic plane) have the same area, namely, Iρ = Aρ (π). Then, using the
not-yet-proved fact that Iρ = πρ2 , (ii) conclude that Aρ (α) = αρ2 . Finally, working from
Figure 7.3, p. 92, (iii) deduce a general formula for the area of a (non-ideal) hyperbolic
triangle.
7.3 Next, we compare results regarding the angle sum of a triangle on different surfaces.
(a) Looking at triangles on a sphere or hyperbolic plane, and remembering your results for
their areas in the last two problems, what can you say about the sum of the interior angles
of a triangle on a sphere or hyperbolic plane? Can they vary? What are their maximum
and minimum values?
(b) What is the sum of the interior angles of a planar triangle? Can you use your work in
part (a) to prove this?
7.4 Suppose that r and r0 are geodesic lines or line segments that intersect a common geodesic l
at respective points P and P 0 . If the corresponding angles made by r with l at P and r0 with l
at P 0 are equal, we say that r and r0 are parallel transports of each other along l. (You should
imagine sliding r along l from P towards P 0 and ending at r0 in such a way as to maintain
the angle made between the lines. See Figure 7.6, p. 96.) Now, consider a vector (in other
words, a directed geodesic segment) emanating from a vertex of some triangle ∆; if you parallel
transport the vector counterclockwise around all three sides of ∆, what is the direction of the
resulting vector? Is it the same as that of the original? The answer is yes if the triangle is
drawn in a plane, but no if the surface is a sphere or hyperbolic plane! Consequently, we call
the difference of these angles (measured counterclockwise on the sphere and clockwise on a
hyperbolic plane) the holonomy of the corresponding triangle and denote this quantity H(∆).
(a) Find a formula that expresses the holonomy of a small triangle on a sphere in terms of
its angle sum, and another in terms of its area. Discuss where your formulas come from.
(See Figures 7.7, p. 98, and 7.9, p. 101, and the discussion at the bottom of p. 100.)
(b) Find a formula that expresses the holonomy of a triangle on a hyperbolic plane in terms
of its angle sum, and another in terms of its area. Discuss where your formulas come
from. (See Figure 7.8, p. 99.)
There are striking similarities in these two formulas. In particular, in both cases the ratio
κ = H(∆)/Area(∆) turns out to be a simple function f (ρ) of ρ, the radius of the surface.
This quantity is commonly called the (Gaussian) curvature of the surface, and the resulting
relation,
H(∆) = κ · Area(∆),
is known as the Gauss-Bonnet Formula for triangles on a surface.
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