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This is the Past Exam Paper of Math Tripos which includes Restriction and Kakeya Phenomena, Minkowski Dimensions, Real Numbers, Besicovitch Set, Besicovitch Subset, Absolute Constant, Measurable Function, Discrete Fourier Analysis etc. Key important points are: Global Riemannian Geometry, Jacobi Field, Conjugate Point, Multiplicity of Conjugate Point, Parametrized Surface, Local Isometries, Constant Sectional Curvature, Induced Metric, Polynomial Growth of Degree
Typology: Exams
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Monday 2 June 2003 1.30 to 4.
Attempt THREE questions.
There are five questions in total. The questions carry equal weight.
1 (a) Let M be a Riemannian manifold. Define Jacobi field, conjugate point and the mutiplicity of a conjugate point.
(b) Show that the multiplicity of a conjugate point cannot exceed n − 1, where n is the dimension of M.
(c) Let M be a Riemannian manifold, γ : [0, 1] → M a geodesic and J a Jacobi field along γ. Prove that there exists a parametrized surface f (t, s) such that f (0, t) = γ(t), the curves t 7 → f (s, t) are geodesics and J(t)= ∂f∂s (t, 0).
2 (a) Let fi : M → N , i = 1, 2 be two local isometries between connected Riemannian manifolds. Show that if there exists p ∈ M such that f 1 (p) = f 2 (p) and (df 1 )p = (df 2 )p, then f 1 (q) = f 2 (q) for all q ∈ M.
(b) Let M be a complete Riemannian manifold with constant sectional curvature K ≡ 1. Show that the universal covering of M with the induced metric is isometric to Sn. [You may assume Cartan’s theorem.]
(c) Let M be an even dimensional complete Riemannian manifold with constant sectional curvature K ≡ 1. Show that M is isometric to Sn^ or to the real projective space of dimension n. Is the same result true in odd dimensions?
3 (a) Let γ : [0, ∞) → M be a unit speed geodesic of a complete Riemannian manifold M. Show that γ cannot minimize distance past its first conjugate point. [You may assume the formula for the second variation of energy.]
(b) Let γ(t 0 ) be the cut point of p = γ(0) along γ. Show that either γ(t 0 ) is the first conjugate point of p along γ or there exists a geodesic σ different from γ, joining p to γ(t 0 ), and such that σ also minimizes the distance from p to γ(t 0 ).
(c) Give an example of a positively curved manifold M and a point p ∈ M such that for any geodesic γ with γ(0) = p, the cut point of p along γ does not coincide with the first conjugate point of p along γ.
4 Let M n^ be a complete manifold with Ric > k > 0.
(a) State the Bonnet-Myers theorem.
(b) Let Skn be the n-dimensional sphere with constant curvature k. Using the classification theorem of complete manifolds of constant sectional curvature, show that if Vol(M ) = Vol(Skn ), then M is isometric to Snk.
(c) Show that if diam(M ) = π/
k, then M is isometric to Skn.
[You may use comparison results for Ricci curvature, but they should all be clearly stated.]
Paper 16
