Announcements

We are meeting in Math 0103 starting Monday, August 26.

Office hours Mondays 2-2:50, or by appointment, in 4412.

No class Monday, December 9. Instead, we will have the final lecture Wednesday, December 11 at 10:30 in our usual room, 0103.

Problem Sets

• HW 1, due 9/11 FINAL
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  1. Let φ be a coordinate chart on an n-dimensional manifold M defined at a point p. Define tangent vectors by $$ v_i(f) = \frac{\partial (f \circ \varphi^{-1})}{\partial x_i} \qquad f \in C^\infty (M) $$ Show that {v₁, ..., v_n} form a basis for T_p M, the tangent space to M at p.
  2. In this problem you are asked to compute tangent spaces of manifolds embedded in Rⁿ⁺¹. In this case, elements of T_p M can equivalently be viewed as vectors in Rⁿ⁺¹, based at p. It is these subspaces of Rⁿ⁺¹ which you are asked to compute below.
     

     Recall the notation for the nondegenerate quadratic forms on Rⁿ of index (p,q), where p+q = n: $$ Q_{p,q}({\bf x}) = x_1^2 + \cdots + x_p^2 - x_{p+1}^2 - x_n^2 $$ Consider the manifolds $$ S^{p,q} = \{ {\bf x} \in {\bf R}^{n+1} \ : Q_{p+1,q}({\bf x}) = 1 \}$$ and $$ H^{p,q} = \{ {\bf x} \in {\bf R}^{n+1} \ : \ Q_{p,q+1}({\bf x}) = -1 \}$$ Show that T_p S^p,q = p^⊥ and similarly for H^p,q. Thus both are pseudo-Riemannian manifolds of index (p,q).

  3. For a smooth path γ:[a,b] → M,
     
     a) assume that M is Riemannian, and let $$ \alpha(t) = \int_a^t \left\| \dot{\gamma}(t) \right\| \ dt$$ Then show that γ composed with α^-1 is parametrized by arc length;
     
     b) assume that M is pseudo-Riemannian, and show that a smooth timelike path admits a parametrization by proper time.
  4. Compute the Euler-Lagrange equations for the hyperbolic plane in the upper-half plane model $$ g_{(x,y)} = \frac{dx^2 + dy^2}{y^2} $$ a) for the energy Lagrangian;
     
     b) for the length Lagrangian.
  5. In lecture we computed the matrix of leading coefficients in the Euler-Lagrange equations for the length Lagrangian on an n-dimensional Riemannian manifold: $$ \left( a_{km} \right) = \left( \frac{g_{km}}{\left\| \dot{\gamma} \right\|} - \frac{\left( \Sigma_{\ell} g_{k\ell} \dot{\gamma}^\ell \right) \left( \Sigma_{\ell} g_{m\ell} \dot{\gamma}^\ell \right) }{\left\| \dot{\gamma} \right\|^3} \right) $$ Show that the rank of this matrix is n-1.

• HW 2, due 9/25 FINAL
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  1. Let γ be a smooth curve in a semi-Riemannian manifold M. Given two coordinate charts $$ \varphi = (x_1, \ldots, x_n) \qquad \bar{\varphi} = (\bar{x}^1, \ldots, \bar{x}^n)$$ use our transformation rules for Christoffel symbols to show that $$ \varphi \circ \gamma = (\gamma^1, \ldots, \gamma^n)$$ satisfies the geodesic equations in the first coordinates if and only if $$ \bar{\varphi} \circ \gamma = (\bar{\gamma}^1, \ldots, \bar{\gamma}^n)$$ satisfies the geodesic equations in the second coordinates.
  2. (Postnikov Exer. 12.3) For the Lagrangian $$ \mathcal{L} = \dot{x}^2 + \dot{y}^2$$ on the R², show that the path x = at, y = 0 is an extremal. Let A be the point (a,0), a > 0, on this extremal, and let O be the origin. The integral $$ \int_0^1 (\dot{x}^2 - \dot{y}^2) dt$$ over the segment OA of this extremal is equal to a². Compute that the action over the broken line OBA (correspondingly parametrized), where B is the point (a/2,b/2), b>0, is equal to a² - b². Therefore, the segment OA is not a minimum curve, and there is no minimum curve connecting the points O and A for this Lagrangian.
  3. Show that a timelike geodesic in a Lorentzian manifold locally maximizes proper time.
  4. In HW 1, you computed the Christoffel symbols for the hyperbolic plane in the upper half-space model: $$ \Gamma_{11}^1 = \Gamma_{22}^1 = \Gamma_{12}^2 = 0 \qquad \Gamma_{11}^2 = - \Gamma_{12}^1 = - \Gamma_{22}^2 = \frac{1}{y}$$
     • Solve for the geodesic γ with γ(0) = i and γ'(0) = i (where i is the complex number, with real coordinates (0,1)).
     • Solve for the parallel transport of an arbitrary vector V ∈ T_i H² along γ.

  5. An isometry of a semi-Riemannian manifold (M, g) is a diffeomorphism f such that for all p ∈ M, the differential D_pf : (T_pM, g_p) → (T_f(p) M, g_f(p)) is a linear isometry. The orientation-preserving isometries of H² in the upper half-space model are the fractional linear transformations $$ f(z) = \frac{az+b}{cz+d} \qquad ad-bc = 1$$ Find a one-parameter subgroup {f^t}_{t ∈ R} of the group of fractional linear transformations such that γ(t) = f^t(i) and the parallel transport P₀^t V of V ∈ T_i H² along γ is D_if^t(V).

• HW 3, due 10/16 FINAL
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  1. Let M = S^p,q or H^p,q, a hypersurface in Rⁿ⁺¹. Recall that you proved for these manifolds that T_xM = x^⊥, where the orthogonal is taken with respect to Q_p+1,q or Q_p,q+1, respectively. A vector field Y on M can be viewed as a function M → Rⁿ⁺¹. Show that for vector fields X and Y on M, ∇ defined by $$ \nabla_X Y(x) = \pi_{x^\perp} ( X(x)(Y) ) $$ where π_x^⊥ is the projection from Rⁿ⁺¹ to x^⊥,
     • satisifies the axioms for a covariant derivative;
     • is symmetric; and
     • is compatible with the metric on M.

     Thus this formula defines the Levi-Civita connection on M.
  2. • Using the previous problem, argue without explicit calculation that the geodesic in M with initial point x and initial velocity u equals, as a set, the intersection of span{x,u} with M. The parametrization is then determined by < γ', γ' > = < u,u >, unless u is lightlike.
     • Show that the lightlike geodesic in M with initial point x and initial tangent vector u is the affine line {x + tu, t ∈ R }

  3. Find the geodesic with initial point x and initial velocity u, with:
     • M = H^2,1, x = (0,0,0,1), and u = (0,0,1,0).
     • M = S^2,1, x = (1,0,0,0), and u = (0,1,0,0).
     • M = H^1,1, x = (0,0,1), u = (1,1,0).

  4. In this problem, you will compute the curvature of S^p,q, at a particular point x = (1,0,...,0).
     • Define n vector fields on a neighborhood of x which evaluate to ∂₁, ..., ∂_n at x.
     • Use these and the formula for ∇ to compute the curvature tensor R at x.
     • Compute the sectional curvature of a nondegenerate plane in T_xS^p,q.

  5. Now do the same for H^p,q.
  6. (Postnikov Exer. 27.1) Given a piecewise smooth vector field X along a geodsic γ defined on [0,b], decompose X into X^⊥ and a vector field tangent to γ. Show that $$ \int_0^b \left( \left< \frac{D (X^\perp)}{dt}, \frac{D (X^\perp)}{dt} \right> - \left< R( X^\perp, \dot{\gamma})\dot{\gamma}, X^\perp \right> \right) dt \leq \int_0^b \left( \left< \frac{D X}{dt}, \frac{DX}{dt} \right> - \left< R( X, \dot{\gamma})\dot{\gamma}, X \right> \right) dt $$

• HW 4, due 10/30 FINAL
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  1. Let f be an isometry of a semi-Riemannian manifold (M,g).
     • Let ∇ be the Levi-Civita connection of (M,g). Show $$ f_*(\nabla_X Y) = \nabla_{f_*X} f_*Y \qquad \forall X,Y \in \mathcal{X}(M)$$ where $$ f_* X (f(p)) = D_pf (X(p))$$
     • For the curvature tensor R, show that $$ f_* (R(X,Y)Z ) = R(f_* X, f_* Y) f_* Z \qquad \forall X,Y,Z \in \mathcal{X}(M)$$
     • Show that if γ is a geodesic, then f compose γ is a geodesic.
  2. Show that in a Lorentzian manifold of nonnegative sectional curvature, given a timelike geodesic γ with initial point p, there are no conjugate points to p along γ.
  3. Let S be the paraboloid z = x² + y² with the Riemannian metric induced from R³.
     • There is only one sectional curvature of a surface, called the Gaussian curvature. Show that it is positive at every point.
     • Clearly S is noncompact. Explain why Mayers' Theorem does not apply to S.
  4. You saw in the previous homework that S^p,q has constant positive sectional curvature and H^p,q has constant negative sectional curvature. Denote this constant sectional curvature by C. Then the Jacobi equation along a geodesic γ in these spaces takes the form $$ \frac{D^2 J}{dt^2} = - C \left< \dot{\gamma}, \dot{\gamma} \right> J$$
     • Show that if γ is spacelike and parametrized by arc length, and if C > 0, then γ(π/|C|^1/2) is conjugate to γ(0). Show that if γ is timelike and parametrized by proper time, and if C < 0, the γ(π/|C|^1/2) is conjugate to γ(0).
     • Show that for any other combination of the type of γ and the sign of C (including for γ lightlike), there are no conjugate points to γ(0) along γ. (You can of course use your work from problem 2.)
• HW 5, due 11/13 FINAL
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  1. Let η : Ω → Rⁿ be a smooth function vanishing on the boundary of Ω, and let L be a Lagrangian on surfaces in Rⁿ. Show that $$ \int\int_{\Omega} \left( \frac{\partial L}{\partial \sigma} - \frac{\partial}{\partial \sigma_u} \frac{\partial L}{\partial \sigma_u} - \frac{\partial}{\partial v} \frac{\partial L}{\partial \sigma_v} \right) \cdot \eta \ du dv = 0$$ for all such η if and only if $$ \frac{\partial L}{\partial \sigma} - \frac{\partial}{\partial \sigma_u} \frac{\partial L}{\partial \sigma_u} - \frac{\partial}{\partial v} \frac{\partial L}{\partial \sigma_v}$$ vanishes identically on Ω
  2. Show that the coordinates $$ u = \theta \qquad v = \ln \tan\left( \phi/2 + \pi/4 \right)$$ are isothermal on S², where θ, φ are the usual spherical "coordinates" corresponding to latitude and longitude, with 0 ≤ θ < 2 π, -π/2 ≤ φ ≤ π/2.

  p

  3. (Postnikov Exer. 14.4) Show that the Enneper surface in R³, with parametrization $$ \sigma(u,v) = (u(3+3v^2 - u^2),-v(3+3u^2-v^2), 3(v^2-u^2))$$ is a minimal surface---that is, it has vanishing mean curvature.
  4. (Nishikawa Exer. 3.10) Let σ : Ω → Rⁿ be a parametrization of an immersed surface, so Ω is a region in R². We have the area functional: $$ A(\sigma) = \int\int_\Omega \left( \langle \sigma_u, \sigma_u \rangle \langle \sigma_v, \sigma_v \rangle - \langle \sigma_u, \sigma_v \rangle^2 \right)^{1/2} du dv$$ and the energy functional $$ E(\sigma) = \int\int_\Omega \frac{\langle \sigma_u, \sigma_u \rangle + \langle \sigma_v, \sigma_v \rangle}{2} du dv$$ Show that A(σ) ≤ E(σ), with equality if and only if σ is an isothermal parametrization.

• HW 6, due 12/4 FINAL
  1. (Nishikawa Exer. 3.8) Regard R² = C by identifying (x,y) with z = x + iy. Express the three-sphere S³ by $$ {\bf S}^3 = \{ (z_1,z_2) \in {\bf C}^2 \ : \ |z_1|^2 + |z_2|^2 = 1 \}$$ Define the Hopf map φ : S³ → S² by $$ \varphi(z_1,z_2) = (2 z_1 \bar{z_2}, |z_1|^2 - |z_2|^2) \in {\bf C} \times {\bf R}, \qquad (z_1, z_2) \in {\bf S}^3$$ Show that φ is a harmonic map with respect to the Riemannian metrics on S³ and S² induced from their standard embeddings in Euclidean space.
  2. (Nishikawa Exer. 4.2) Prove the Weitzenböck formula for κ: $$ \frac{\partial \kappa(u_t)}{\partial t} = \Delta \kappa(u_t) - \left| \nabla \frac{\partial u_t}{\partial t} \right|^2 + \sum_{i=1}^m \langle R^N \left( du_t(X_i),\frac{\partial u_t}{\partial t} \right) \frac{\partial u_t}{\partial t}, du_t(X_i) \rangle $$ for { X₁, ... , X_m } a (local) orthonormal framing of M.
  3. (Nishikawa Exer. 4.4) Given maps f: L → M and h: M → N, between Riemannian manifolds, show that the second fundamental form of h compose f satisfies $$ \nabla d (h \circ f) = \nabla dh (df,df) + dh \circ \nabla df$$ and the tension fields satisfy $$ \tau( h \circ f) = \mbox{tr}^L \nabla dh (df,df) + dh \circ \tau(f)$$