Announcements

Office hours Wednesday 3-4:30, or by appointment, in 4103 starting September 7.

Homework problems

1. Let Γ be a group acting smoothly, freely, and properly discontinuously on a manifold M. Prove that Γ \ M is a smooth manifold such that the quotient map q : M → Γ \ M is smooth. You may use without proof that q is a covering map.
2. Let φ : Sⁿ → Rⁿ be the stereographic projection from the north pole.
   (a) Compute φ^-1 as a map Rⁿ → Rⁿ⁺¹.
   (b) Compute ∂/∂x₁(p) for p not equal the north pole.
   (c) Sketch the resulting vector field on the sphere.
3. Let p be a point of a manifold M. Find a way to associate to each curve through p a tangent vector at p. Show that all v ∈ T_p M can be obtained in this way.
4. (a) Show that for any manifold M, D(Id_M)_p is the identity on T_pM for all p ∈ M.
   (b) Suppose f is a diffeomorphism M → N. Show that for all p, Df_p is an isomorphism with inverse Df^-1_f(p).

for 9/23 FINAL

1. Compute the coefficient functions &sigma_ij for the standard metric on S² in the two coordinate charts given by stereographic projection (x,y,z) &rarr (x/(1+z),y/(1+z)) or (x/(1 - z),y/(1 - z)).
2. Consider the vector fields X = ∂/∂x and Y = ∂/∂y + x ∂/∂z on R³.

   (a) Compute the flows along each of X and Y.

   (b) Compute the bracket [X,Y].

   (c) Show that any point of R³ can be reached by flowing along X or Y for a finite sequence of times.

3. Show that the vector fields X = y²∂/∂x and Y = x²∂/∂y on R²are complete, but X + Y is not.
4. Let f be a smooth map between manifolds M^m and Nⁿ, and let p &isin M.
   (a) Suppose f is an immersion at p. Show there exist charts (&phi,U) at p and (&psi,V) at f(p) such that the composition &psi &sdot f &sdot &phi^-1, from &phi(U) &sube R^m to &psi(V) &sube Rⁿ, is (u₁, . . . , u_m) &rarr (u₁, . . . , u_m, 0 , . . . , 0) (b) Suppose f is a submersion at p. Show there exist charts (&phi,U) at p and (&psi,V) at f(p) such that the composition &psi &sdot f &sdot &phi^-1, from &phi(U) &sube R^m to &psi(V) &sube Rⁿ, is (u₁, . . . , u_m) &rarr (u_m-n+1, . . . , u_m)
5. Let D be a k-dimensional involutive distribution on Mⁿ. Given a point p, the leaf L_p comprises all the points of M that can be joined to p by flowing along a finite sequence of vector fields in D. Show that L_p is a path-connected, k-dimensional manifold containing p and tangent to D (that is, every smooth path γ : (- ε, ε) &rarr L_p gives a smooth path in M with γ'(0) in D). Show that L_p is maximal with the above properties. (Hint: Given a point q ∈ L_p, take a foliated chart at q.)

6. (O'Neill Ex. 1.14) A critical point of a smooth function f : M → R is p ∈ M where Df_p =0.

   (a) At such a point there exists a Hessian H : T_pM × T_pM → R defined by H(X_p,Y_p) = X_p(Yf). Verify that this definition only depends on Y_p and that H is symmetric. (Hint: Take a coordinate chart, and express in terms of coordinate vector fields.)

   (b) Show H(v,v) = (∂²(f &sdot γ)/∂t²)(0) if γ is a path with γ'(0) = v.

for 10/7 FINAL

1. Compute the exponentials of the following elements of the Lie algebra sl(2,R):
   1. (0 1)
      (0 0)
   2. (1 0)
      (0-1)
   3. (0 1)
      (-10)
   Now show that
   (-1 1)
   (0 -1)
   is not in the image of the exponential map sl(2,R) → SL(2,R.).

2. Compute T₁O(p,q), and verify it is closed under bracket.

3. Consider the Heisenberg group with underlying manifold R³ defined in class. The identity element is (0,0,0). Compute the left-invariant vector fields X,Y, and Z, evaluating to the standard basis vectors e₁, e₂, and e₃, respectively, at the origin. Now compute the Lie algebra by computing the brackets of these vector fields. Show that the one-parameter subgroups e^tX and e^tY generate the group.

4. Let G be a connected Lie group with Lie algebra g.

   (a) Show that G is abelian if and only if g is abelian.

   (b) Show that if h is an ideal of g, then the connected subgroup H < G tangent to h is normal. Conversely, if H is a normal subgroup of G, then the Lie algebra h of H is an ideal in g.

5. Show that a smooth vector bundle over a smoothly contractible manifold is smoothly trivializable. (The same statement holds in the continuous category.)
   HERE IS A NICE HANDOUT by N. Nowaczyk CONTAINING A SOLUTION.

6. Let ρ : Γ → V be a representation of Γ ≅ π₁(N). Show that there is a 1-1 correspondence between sections of Ñ ×_ρ V and ρ-equivariant functions from Ñ to V. Show that this correspondence is actually an isomorphism of C^∞(N)-modules.

1. In class we computed the parallel transport along a great circle in the sphere, where the great circle is the intersection of Sⁿ with the plane spanned by two unit vectors p and q with p ⊥ q. Assume these paths are parametrized with unit speed, so they are 2π-periodic curves, and denote by γ^1/4_p,q this path restricted to [0,&pi/2].

   (a) Let p, q, and r be mutually orthogonal unit vectors in Rⁿ⁺¹. Compute the parallel transport along the concatenation γ_p,q^1/4 followed by γ_q,r^1/4 followed by γ_r,p^1/4.

   (b) Compute the holonomy group of the standard connection on Sⁿ at an arbitrary point p. (Hint: Use that Hol_p is contained in the orthogonal group of T_p Sⁿ, which is isomorphic to O(n).

2. Let &Gamma_ij^k be the Christoffel symbols for a connection on the tangent bundle TM in a coordinate system &phi = (x¹, . . ., xⁿ). Let ψ = (y¹, . . ., yⁿ) be another coordinate chart on the same domain. Express the Christoffel symbols in this new chart in terms of the Γ_ij^ks and the transition map.

3. Let Γ ≅ π₁(N) and ρ : &Gamma → GL(V) be a representation, and consider the connection defined in class on the bundle E = Ñ ×_ρ V. Compute the holonomy of this connection at a point of N.

4. Let H be a horizontal distribution on a vector bundle E satisfying the two axioms from class. Let ρ: TE → VE be the projection determined by H.

   (a) Verify that for &nu ∈ E and c ∈ R,

   ρ_cν (μ_c)* = (μ_c)* &rho_ν
   where μ_c is scalar multiplication by c on E.

   (b) Verify that the covariant derivative defined by

   (&nabla_X &sigma)_p = t_σ(p) ρ_σ(p) (X.σ)_p
   is C^∞(M)-linear in X and satisfies the Leibniz axiom in σ (don't worry about additivity in σ.).

5. Let B be a principal G-bundle, for G a Lie group. Show that the right action of G on B is proper.

6. Let E = TM for a manifold M, and let B be the bundle of frames of E. (This is called the frame bundle of M). It is a principal GL(Rⁿ)-bundle. Suppose that B' ⊂ B is a principal O(n)-bundle over M, and let ρ be the standard representation of O(n). Now define a Riemannian metric on the associated bundle B ×_ρ Rⁿ. Conversely, given a Riemannian metric on M, how can you produce a principal O(n)-bundle on M?

1. Compute the Christoffel symbols for the standard connection on R² in polar coordinates.

2. (a) Show that the orthonormal frame bundle of Sⁿ can be identified with the group O(n+1).

   (b) More generally, express O(p+1,q) as a principal O(p,q)-bundle over S^p,q. What is the interpretation of this bundle in terms of the pseudo-Riemannian metric on the base?

3. (a) Show all geodesics γ in S^p,q are contained in P &cap S^p,q, where P is the plane spanned by x = &gamma(0) and v₀ = &gamma'(0), and similarly for H^p,q.
   (b) Compute the geodesics of H^n,0, hyperbolic space in the hyperboloid model, in terms of the initial conditions &gamma(0) = x₀ and &gamma'(0) = v₀, assuming <v₀,v₀> = 1.
   

4. Compute the distance between two points v,w of hyperbolic space in the hyperboloid model, H^n,0, in terms of <v,w>.

5. Let α be a curve in a Riemannian manifold defined on an interval I with <α'(t),α'(t)> nonzero for all t ∈ I. Show that there is a reparametrization c of α with constant speed---so that <(α c)', (α c)'> is constant.

6. (O'Neill problem 3.19) A curve is called a pregeodesic if some reparametrization of it is a geodesic. Suppose that α is a curve in a semi-Riemannian manifold with α' never 0 and Dα'/dt collinear with &alpha'. Show that α is a pregeodesic with the following steps.

   (a) Write Dα'/dt (t) = f(t)⋅α'. Then a reparametrization α(c) is a geodesic if and only if c'' + f(c)⋅(c')² = 0.

   (b) If <α',α'> is never 0, then any constant speed reparametrization of α is a geodesic.

   (c) <α',α'> is always zero or never 0.

   (d) α is a pregeodesic in the case <α',α'> = 0.

1. (O'Neill problem 3.18) The curl of a vector field V on M is defined by
   (curl V)(X,Y) = < &nabla_X V,Y> - <∇_YV,X>
   

   (a) Show that that curl V is skew-symmetric and bilinear over C^∞(M), with coordinate components ∂V_j/∂xⁱ - ∂V_i/ ∂x^j.

   (b) Show curl(grad f) = 0.

   (c) On R³, (curl V)(X,Y) = (X × Y) • (∇ × V), where ∇ = (&part/∂x, ∂/∂y, &part/∂z).

2. (the real Hessian) For a smooth function f on M, define the Hessian H^f(X,Y) = XY(f) - (∇_XY)(f). Note that H^f equals XY(f) at a critical point of f. Show that H^f(X,Y) = <∇_X(grad f),Y> and that H^f is symmetric.

3. (a) Show that a convex combination t∇ + (1-t)&nabla' of connections is a connection.

   (b) Use partitions of unity to show that any vector bundle admits a connection.

4. (O'Neill 5.12) In a complete Riemannian manifold

   (a) If &sigma_i are minimizing geodesics on [0,1] and σ_i'(0) converges to v ∈ T_p M, then &sigma_v is minimizing.

   (b) If M is not compact, then there exists a minimizing ray &sigma : [0,∞) → M starting at p (that is, each subsegment of &sigma is minimizing).

5. (O'Neill 5.13 (a)) Call a semi-Riemannian manifold M Misner-complete provided no geodesic races to infinity---that is, provided every geodesic &gamma : [0,b) → M, where b < ∞, lies in a compact set. Prove geodesically complete implies Misner complete implies M does not isometrically embed as an open submanifold of a strictly larger connected semi-Riemannian manifold.

for 12/9

1. (O'Neill 9.12) Let T be the Clifton-Pohl torus defined in class.

   (a) Find 4 isometries of T.

   (b) Show that &gamma(t) = (tan t,1) is a geodesic, and deduce that every null geodesic of T is incomplete.

   (c) The flow along u ∂/∂u + v ∂/∂v for any time is an isometry.

   (d) If &gamma(t) = (u(t),v(t)) is a geodesic, then both u'v'/r² and (uv'+vu')/r² are constant, where r² = u² + v².

   (e) T is not geodesically connected.

   (f) &gamma(t) = (1,1/t) is a pregeodesic with finite length on [0,∞).

   (g) T has timelike geodesics that are complete and ones that are not complete; same for spacelike.

2. Let P : Ω^k(R × M) → Ω^k-1(M) be the operator on forms defined in class. Let &pi : R × M → M and i_a : M → R × M, i_a(x) = (a,x) be as defined in class. Prove the formula dPω + Pd&omega = ω - &pi* i_a* ω.

3. (Guillemin and Pollack 7.4) Derive the classical Stokes' Theorem: Let S be a compact oriented 2-manifold in R with boundary, and let F = (f₁, f₂, f₃) be a smooth vector field in a neighborhood of S. Prove
   ∫_S (curl F ⋅ n) dA = ∫_∂S (f₁ dx₁ + f₂ dx₂ +f₃ dx₃)
   where n is the outward normal to S and dA is as defined in class.

4. (Guillemin and Pollack 7.8) Suppose that X = ∂W, W is compact, and f: X → Y is a smooth map. Let ω be a closed k-form on Y, where k = dim X. Prove that if f extends to all of W, then ∫_X f^*ω = 0.

5. (Guillemin and Pollack 7.9) Suppose that f₀, f₁ : X → Y are homotopic maps and that the compact, boundaryless manifold X has dimension k. Prove that for all closed k-forms ω on Y, ∫_X f₀^*ω = ∫_X f₁^*ω

6. (Guillemin and Pollack 7.10) Show that if X is a simply connected manifold, then ∫_γ ω = 0 for all closed 1-forms ω on X and all loops γ in X.