Riemannian and Symplectic Geometry

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Let $M$ be a Riemannian manifold, $exp_p:T_pM \to M$ the exponential map at a point $p$ . Is it true that for each $q \in T_pM$ the exponential map is locally a diffeomorphism around $q$ ?
List all the Jacobi fields along a great circle joining the north and south poles on $S n$ . (Weinstein)
Find two equal-volume flat tori that are not isometric.
Compute the curvature of the unit sphere in $\R^3$ .
What is parallel translation? How is it related to the notion of a connection on a principal bundle?
Given a principal $O (n)$ bundle, give some examples of associated vector bundles.
Give $S 2$ the usual induced metric from $\R^3$ and the (local) parameterization $(\theta,\phi) \to (\cos \phi \cos \theta, \cos \phi \sin \theta, \sin \phi)$ Construct an orthonormal basis for $T * S 2$ . Calculate the Levi-Civita connection with respect to this basis. Calculate the Christoffel symbols. Calculate the curvature and the first Chern class.
What are the geodesics on $\R^n$ ? On $S 2$ ? On the hyperbolic plane?
Derive the Euler-Lagrange equations.
Prove that a closed surface in $\R^3$ cannot have everywhere negative gaussian curvature.
Let $M$ contain a totally geodesic surface $S$ . What can you say about the curvature of $S$ ?
Give an example of a vector field in $\R^2$ which is not uniquely integrable.
A Riemannian metric naturally identifies $T X$ and $T * X$ . What kind of form does a volume-preserving flow correspond to?
What is the area of a right-angled hyperbolic heptagon? What is the smallest genus closed hyperbolic surface that can be decomposed into right-angled hyperbolic heptagons?
What is the condition on a $1$ -form $α$ on a three manifold that $\ker(\alpha)$ be integrable?
Is the space of Riemannian metrics on $M$ path connected, for a fixed smooth structure? Is it contractible for $M = S 1$ ? What about $M = S 2$ ?
What is the exponential map for a Riemannian manifold? (Pugh)
What is a geodesic? Do they always exist? Are they unique? What are the Christoffel symbols? (Pugh)
Let $γ v (t)$ denote the geodesic through $p \in M$ with initial tangent vector $v \in T_pM$ . Why is $γ t v (1) = γ v (t)$ ? (Pugh)
What does it mean for two points $p,q \in M$ to be conjugate? (Weinstein)
If any two points in a Riemannian manifold can be joined by a minimizing geodesic, does that mean the manifold is geodesically complete? Give some examples of incomplete manifolds. (Weinstein)
Let $S$ be a surface homeomorphic to $S 2$ . Let $p, q$ be two points on $S$ and $γ$ a minimizing geodesic connecting them. Prove that there must exist at least one more geodesic connecting $p$ and $q$ . (Weinstein)
What is a symplectic manifold? Why must the dimension be even? List all the symplectic manifolds you know. (!) (Weinstein)
Why is $S^1 \times S^3$ not symplectic? What about ${\mathbb C} P^2 \# {\mathbb C} P^2$ ?
Give an example of a symplectic group action which is not hamiltonian.
Give an example of a hamiltonian group action which is not Poisson.
Given an example of a non-integrable almost-complex structure.
Calculate the moment map for the standard action of $SO(3,\R)$ on $\R^3$ .
Calculate the symplectic volume of $S^{2n+1}_{\sqrt{2E}}$ as a function of $E$ , where this denotes the inverse image of $2 E$ under the moment map $\mu:(z_0, \dots, z_n)) \to 1/2(|z_0|^2 + \dots + |z_n|^2)$ for the standard $S 1$ action on ${\mathbb C}^{n+1}$ . (Weinstein)
Give a counterexample to Moser's theorem if $M$ is not compact.
Explain geodesics to a symplectic geometer.
Let $(M,ω)$ be symplectic where $ω$ is actually an integral class. Can you find a principal $S 1$ bundle $P$ over $M$ and a connection form $θ$ on $P$ such that the curvature of $θ$ is precisely $ω$ ?
Let $P$ and $θ$ be constructed as above. Show that the horizontal distribution on $P$ defined by $θ$ is a contact structure on $P$ .
In the setup above, what is $P$ if $M= {\mathbb C} P^n$ with the canonical symplectic structure? Can you calculate $θ$ in this case? (Weinstein)
Again, let $P,θ$ be as above. Suppose we have another $S 1$ -action on $M$ which lifts to an action of $P$ on $P$ such that the action of the two $S 1$ 's commute and for which the second $S 1$ leaves $θ$ invariant. Is this action hamiltonian (on M)?
Characterize the Lagrangian submanifolds of $T * X$ with the canonical symplectic structure which are "close" to the $0$ -section.
Let $(M,ω)$ be symplectic, $X$ a submanifold defined as the intersection of the $0$ -levels of functions $f_1,\dots,f_k:M \to \R$ . (Suppose $0$ is a regular value for each $f i$ ). Suppose each $T x X$ is coisotropic. What can you say about $X$ ?
Let $(M,ω)$ and $X$ be as above, but now suppose that $T x X$ is a symplectic subspace of $T x M$ for each $x \in X$ . What can you say about $X$ ?
Given $(M,ω)$ symplectic, why is the space of compatible almost-complex structures contractible, and what is this fact good for?
What is a contact manifold? Give some examples of contact manifolds. (Weinstein)
What is a moment map? What is symplectic reduction? Give an example. (Weinstein)
What is the Duistermaat-Heckman theorem? Give an example of its use. (Borcherds)
What does a Hamiltonian vector field look like in local coordinates? (Halpern)
What are the geodesics on a Lie group with biinvariant metric? (Weinstein)
Does the Lie group $S U (2)$ admit a metric with constant curvature? Can you prove this using some transitive isometric group action on it? (Weinstein)
Let $M$ be a compact toric variety with the effective action of a torus of half the dimension of $M$ . What can you say about the actions of a subtorus $S$ ? What degree will the Duistermaat-Heckman polynomials for the $S$ -action have? Will the action on the level sets of the $S$ -moment-map be almost free? (Knutsen)
Formulate the main difference between Riemannian and symplectic geometry in one sentence. (S. Dyatlov vs. Zworski)