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 22:50, 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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 Let φ be a coordinate chart on an ndimensional 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 {v_{1}, ..., v_{n}} form a basis for T_{p} M, the tangent space to M at p.
 In this problem you are asked to compute tangent spaces of manifolds embedded in R^{n+1}. In this case, elements of T_{p} M can equivalently be viewed as vectors in R^{n+1}, based at p. It is these subspaces of R^{n+1} which you are asked to compute below.
Recall the notation for the nondegenerate quadratic forms on R^{n} 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 pseudoRiemannian manifolds of index (p,q).
 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 pseudoRiemannian, and show that a smooth timelike path admits a parametrization by proper time.
 Compute the EulerLagrange equations for the hyperbolic plane in the upperhalf plane model
$$ g_{(x,y)} = \frac{dx^2 + dy^2}{y^2} $$
a) for the energy Lagrangian;
b) for the length Lagrangian.
 In lecture we computed the matrix of leading coefficients in
the EulerLagrange equations for the length Lagrangian on an
ndimensional 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 n1.
 HW 2, due 9/25 FINAL
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 Let γ be a smooth curve in a semiRiemannian 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.
 (Postnikov Exer. 12.3) For the Lagrangian
$$ \mathcal{L} = \dot{x}^2 + \dot{y}^2$$
on the R^{2}, 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 a^{2}.
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 a^{2}  b^{2}. Therefore, the
segment OA is not a minimum curve, and there is no minimum curve
connecting the points O and A for this Lagrangian.
 Show that a timelike geodesic in a Lorentzian manifold
locally maximizes proper time.
 In HW 1, you computed the Christoffel symbols for the
hyperbolic plane in the upper halfspace 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} H^{2} along γ.
 An isometry of a semiRiemannian manifold (M, g) is a
diffeomorphism f such that for all p ∈ M, the differential
D_{p}f : (T_{p}M, g_{p}) →
(T_{f(p)} M, g_{f(p)}) is a linear isometry. The
orientationpreserving isometries of H^{2} in the
upper halfspace model are the fractional linear
transformations
$$ f(z) = \frac{az+b}{cz+d} \qquad adbc = 1$$
Find a oneparameter subgroup {f^{t}}_{t ∈
R} of the group of fractional linear transformations such
that γ(t) = f^{t}(i) and the parallel transport
P_{0}^{t} V of V ∈ T_{i}
H^{2} along γ is
D_{i}f^{t}(V).
 HW 3, due 10/16 FINAL
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 Let M = S^{p,q} or H^{p,q}, a hypersurface
in R^{n+1}. Recall that you proved for these
manifolds that T_{x}M = 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 → R^{n+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
R^{n+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 LeviCivita 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 }
 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).
 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_{x}S^{p,q}.
 Now do the same for H^{p,q}.
 (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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 Let f be an isometry of a semiRiemannian manifold
(M,g).
 Let ∇ be the LeviCivita 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.

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.
 Let S be the paraboloid z = x^{2} + y^{2}
with the Riemannian metric induced from
R^{3}.
 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.

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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 Let η : Ω → R^{n} be a smooth function vanishing on the boundary of Ω, and let L be a Lagrangian on surfaces in R^{n}. 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
 Show that the coordinates
$$ u = \theta \qquad v = \ln \tan\left( \phi/2 + \pi/4 \right)$$
are isothermal on S^{2}, where θ, φ are the usual spherical "coordinates" corresponding to latitude and longitude, with 0 ≤ θ < 2 π, π/2 ≤ φ ≤ π/2.
p
 (Postnikov Exer. 14.4) Show that the Enneper surface in R^{3}, with parametrization
$$ \sigma(u,v) = (u(3+3v^2  u^2),v(3+3u^2v^2), 3(v^2u^2))$$
is a minimal surfacethat is, it has vanishing mean curvature.
 (Nishikawa Exer. 3.10)
Let σ : Ω → R^{n} be a parametrization of an immersed surface, so Ω is a region in R^{2}. 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
 (Nishikawa Exer. 3.8) Regard R^{2} = C by identifying (x,y) with z = x + iy. Express the threesphere S^{3} by
$$ {\bf S}^3 = \{ (z_1,z_2) \in {\bf C}^2 \ : \ z_1^2 + z_2^2 = 1 \}$$
Define the Hopf map φ : S^{3} → S^{2} 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 S^{3} and S^{2} induced from their standard embeddings in Euclidean space.
 (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 { X_{1}, ... , X_{m} } a (local) orthonormal framing of M.
 (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
$$ \tau( h \circ f) = \mbox{tr}^L \nabla dh (df,df) + dh \circ \tau(f)$$