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MATH 782 Differential Geometry : homework assignment five 1. A ray is a geodesic γ : [0, ∞) → M which minimizes the distance from γ(0) to γ(t) for all t ∈ [0, ∞). If M is complete and non-compact, prove that there is a ray starting at γ(0) = p for all p ∈ M . 2. Let M and M be Riemannian manifolds and let f : M → M be a diffeomorphism. Assume that M is complete and that there exists a constant c > 0 such that |v| ≥ c|dfp (v)| for all p ∈ M and all v ∈ Tp M . Prove that M is complete. 3. Let M be the upper half-plane R2+ with the metric ds2 = dx2 + dy 2 . yk For which values of k is M complete? 4. a) Let A= a b c −a ∈ sl(2, R) be an arbitrary matrix in the Lie algebra of SL(2, R), and let d = −detA = a2 + bc. Prove that √ √ ω A if d > 0, where ω = d= a2 + bc, cosh ωI + sinh ω √ √ A e = cos ωI + sinω ω A if d < 0, where ω = −d = −a2 − bc, I +A if d = 0. b) Using part a), prove that the image of the exponential map applied to sl(2, R) is precisely the subset {B ∈ SL(2, R) | traceB > −2 or B = −I} of SL(2, R). 1