Honors Exam in Geometry 2007
1. Let P, Q ∈ H2 be any two distinct points in the hyperbolic plane, and suppose P', Q' ∈ H2 are points such that the line segments PQ and P'Q' have the same length. Show that there are exactly two isometries F: H2 → H2 such that F(P) = P' and F(Q) = Q'.
2. Let S be the surface parametrized by X: U → R3, where
X(u, v)=(u, v, u2 - v2),
U = {(u, v): u2 + v2 < 1}.
(a) Calculate the first fundamental form of S in terms of (u, v).
(b) Calculate the area of S.
3. Let S ⊂ R3 be a regular surface. Prove that S is complete if and only if every subset of S that is closed and bounded (in the intrinsic metric) is compact.
4. Let S ⊂ R3 be a compact, connected regular surface of positive genus. Show that there exist points p1, p2, p3 ∈ S such that K(p1) < 0, K(p2) = 0, and K(p3) > 0, where K denotes Gaussian curvature.
5. Suppose ?: S2 → S2 is a global isometry of the sphere. Show that there exists a linear map A: R3 → R3 such that ? = A|S^2.
6. Let α: (a, b) → R3 be a smooth curve of the form α(t) = (0, f(t), g(t)) with f(t) > 0 for all t ∈ (a, b). Let S be the surface of revolution generated by revolving α about the z-axis. Show that the restriction of α to any closed interval [a0, b0] ⊂ (a, b) is distance-minimizing.
7. Two lines in the hyperbolic plane H2 are said to be asymptotically parallel if they admit unit-speed parametrizations γ1, γ2: R → H2 such that d(γ1(t), γ2(t)) is bounded as t → +∞, or, equivalently, if their representations in the Poincar´e disk model approach the same point on the boundary of the disk. An ideal triangle is a region in H2 whose boundary consists of three distinct lines, each pair of which are asymptotically parallel. Prove that all ideal triangles have the same finite area, and compute it.
8. Consider the following three regular surfaces in R3:
S1 = {(x, y, 0) : x, y ∈ R};
S2 = {(x, y, z) : x2 + y2 = 1, 0
S3 = {(x, y, z) : z = x2 + y2}.
For each surface, answer the following questions: Is it bounded? Is it complete? Is it flat? (A surface is flat if each point has a neighborhood that admits an isometry with an open subset of the Euclidean plane.)