HONORS EXAMINATION IN GEOMETRY, 2013

(1) If C is a closed curve contained inside a disk of radius r in R², prove that there exists a point p ∈ C where the curvature satisfies |κg(p)| ≥ 1/r. What analogous statement is true if C is an n-dimensional compact Riemannian manifold in Rⁿ⁺¹ contained inside an n-dimensional sphere of radius r? Is this result still true if C is a compact Riemannian manifold of any dimension in Rⁿ⁺¹ contained inside an n-dimensional sphere of radius r?

(2) Prove that Sⁿ (the n-dimensional sphere of radius 1) has constant sectional curvature equal to 1.

(3) Consider the sphere and the cylinder:

S² = {(x, y, z) ∈ R³| x² + y² + z² = 1},

C = {(x, y, z) ∈ R³| x² + y² = 1}.

Let f: (S²- {(0, 0, 1),(0, 0, -1)}) → C denote the function that sends each point of the domain to the closest point in C. Prove that f is "equiareal" which means that it takes any region of the domain to a region of the same area in C.

(4) Let M² ⊂ R³ be a ruled surface. This means that for all p ∈ M² there exists a line in R³ through p which is entirely contained in M². Explain why the Gauss curvature of M² is non-positive.

(5) Let M = {(x, y, z) ∈ R³| z = x² + y² - 1}, and let C be the circle along which M intersects the xy-plane.

(a) The parallel transport once around C has the effect of rotating the tangent space T_(1,0,0)M by what angle?

(b) What is the integral of the Gauss curvature over the region of M that is bounded by C?

(6) Let M be a Riemmanian manifold, and let f: M → R be a smooth function. Define gradf to be the unique vector field on M such that for all p ∈ M and all X ∈ T_pM, ((gradf)(p), X) = df_p(X) . If gradf has constant norm on M, prove that all integrals curves of gradf are minimizing geodesics.

(7) Let M be a Riemannian manifold. A "ray" in M is define as a geodesic γ: [0, ∞) → M which is minimizing between any pair of points of its image. Suppose that p ∈ M and {V_n} → V is a convergent sequence of vectors in T_pM. Suppose that for each n, the geodesic in the direction of V_n is a ray. Prove that the geodesic in the direction of their limit, V, is a ray.

(8) A Riemannian manifold M is called "homogeneous" if for every pair p, q ∈ M there exists an isometry f: M → M such that f(p) = q. A Riemannian manifold M is called a "symmetric space" if for every p ∈ M there exists an isometry f: M → M such that f(p) = p and df_p: T_pM → T_pM equals the antipodal map V |→ -V . Prove that every complete symmetric space is homogeneous. What examples of symmetric and homogeneous spaces can you think of?

(9) What can you conclude about a surface M² ⊂ R³ for which the image of the Gauss map is contained in a great circle of S²?

(10) Let Sⁿ(r) denote the n-dimensional sphere of points in Rⁿ⁺¹ at distance r from the origin. Let m, n ≥ 1 be integers and let s, r > 0 be real numbers. The product manifold, M = S^m(r) × Sⁿ(s), can be identified with the following subset of R^m+n+2:

M = {(p, q) ∈ R^m+1 ⊕ Rⁿ⁺¹ ≅ R^m+n+2 | p ∈ S^m(r), q ∈ Sⁿ(s)},

and therefore M inherits a natural "product" metric from the ambient Euclidean space. Under what conditions on {m, n, s, r} (if any) will M have...

(a) ...positive sectional curvature?

(b) ...positive Ricci curvature?

(c) ...positive scalar curvature?

(d) ...constant Ricci curvature?