DIFFERENTIAL GEOMETRY FINAL EXAM
AM 8:30 – 12:30, 1/04, 2013 A COURSE BY CHIN-LUNG WANG
1. (a) For any p ∈ M, show that for the Riemann normal coordinate system (U, x)we haveΓkij(p) = 0 and ∂kgij(p) =0 for all i, j, k.
(b) Prove the second Bianchi identity Rij[kl;m] =0.
(c) Let n≥3. If Rij(x) =λ(x)gij(x), show that λ=R/n must be a constant.
2. (a) Derive the first variation for piece-wise C1curves. Use it to show that if γ : [0,`] → M is a closed piece-wise C1 curve which is a critical point of the length functional then γ is a C∞ closed geodesic.
(b) Derive the second variation formula for geodesics. Use it to prove Synge’s theorem: An even dimensional compact oriented manifold with K > 0 must be simply connected.
3. (a) For a normal variation of closed immersed minimal hypersurface Mm → Mm+1with variation field η = f~n, it is known that
A00(0) = Z
M|∇f|2− (Ric(~n,~n) + ||B||2)f2.
Use it to show that if M3has ¯R>0, there is no stable minimal immersion of orientable surface M with g(M) ≥ 1. (Hint: Show that Ric(~n,~n) =
1
2R¯ −KM−12||B||2.)
(b) Construct an isometrically and minimally embedded flat tori T in S3. Why is this not a counterexample to (a)?
4. (a) Prove the Bochner formula: Let ei ∈ TpM be an ONB and ηi ∈ Tp∗M be its dual basis. Then for ω ∈ Ak(M),
(4ω)(p) = −tr∇2ω−∑
i,j
ηi∧ιejR(ei, ej)ω.
(b) If M is compact with Rij ≥ 0, show that h1(M) ≤ dim M. Moreover if Ric(p) >0 for some p ∈ M then h1(M) =0.
5. (a) Let G be a compact Lie group. Construct a bi-invariant metrich,ion G.
(b) Compute∇XY and R(X, Y)Z for X, Y, Z ∈ g, and prove K ≥ 0. When does G have Ric >0? K >0? Apply your answers to U(n)and SU(n). The Bonus problem is on the next page:
1
2 NTU DG FINAL 2013
6. Give the details of one and only one problem in the following list:
(i) Prove the Gauss Lemma, the local minimality of arc length along geodesics, and the existence of convex neighborhoods.
(ii) State and prove the Hopf-Rinow Theorem.
(iii) Define Jacobi fields along a geodesic. For M a complete Riemannian manifold and p ∈ M, show that d expp is singular at v if and only if there is a Jacobi field J 6≡ 0 along γ(t) = expp(tv)with J(0) =0 = J(1). Use it to prove the Cartan–Hadamard Theorem.
(iv) Derive the second variation formula for immersed minimal submanifods and then deduce from it the formula for A00(0)in 3. (a).
(v) Define Riemannian submersion. Derive the formula for the Levi-Civita connection and O’Neil’s curvature formula. Apply it to compute the sec- tional curvature ofCPn.
(vi) Define L2Sobolev spaces Hs(Rm), s∈ R. Prove the Sobolev Lemma and the Rellich Lemma. Extend all these to a vector bundle over a compact manifold.
(vii) Use Garding’s inequality to prove the Hodge decomposition theorem on Ak(M).
