### MATH 742 : Geometric Analysis

Fall 2019
Here is the syllabus for the course in pdf format.

#### 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, ..., vn} form a basis for Tp 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 Tp 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 Tp Sp,q = p and similarly for Hp,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 ∈ Ti 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 Dpf : (TpM, gp) → (Tf(p) M, gf(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 {ft}t ∈ R of the group of fractional linear transformations such that γ(t) = ft(i) and the parallel transport P0t V of V ∈ Ti H2 along γ is Dift(V).
• HW 3, due 10/16 FINAL

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1. Let M = Sp,q or Hp,q, a hypersurface in Rn+1. Recall that you proved for these manifolds that TxM = x, where the orthogonal is taken with respect to Qp+1,q or Qp,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 πx 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.
• 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 }

2. Find the geodesic with initial point x and initial velocity u, with:
• M = H2,1, x = (0,0,0,1), and u = (0,0,1,0).
• M = S2,1, x = (1,0,0,0), and u = (0,1,0,0).
• M = H1,1, x = (0,0,1), u = (1,1,0).
3. In this problem, you will compute the curvature of Sp,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 TxSp,q.

4. Now do the same for Hp,q.
5. (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 Sp,q has constant positive sectional curvature and Hp,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.
3. p

4. (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.
5. (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 : S3S2 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, ... , Xm } 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 $$\tau( h \circ f) = \mbox{tr}^L \nabla dh (df,df) + dh \circ \tau(f)$$