伯克利资格考试与口试:几何
\title{Sample Questions from Past Qualifying Exams}
This list may give the impression that the exams consist of a series of questions fired at the student one after another. In fact most exams have more the character of a conversation with considerable give and take. Hence this list cannot be expected to indicate accurately the difficulties involved.
The list indicates the professor associated to each question where available. Some have been in the MGSA files for a while, and this information has been lost (if it was ever there).
The listing by section is approximate, since some questions may fit under more than one heading.
\section{Geometry, Riemannian and Symplectic}
\begin{enumerate}
\item 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 $q$?
\item List all the Jacobi fields along a great circle joining the north and south poles on $S^n$[\emph{Weinstein}]
\item Find two equal-volume flat tori that are not isometric.
\item Compute the curvature of the unit sphere in $\R^3$.
\item What is parallel translation? How is it related to the notion of a connection on a principal bundle?
\item Given a principal $O(n)$ bundle, give some examples of associated vector bundles.
\item 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.
\item What are the geodesics on $\R^n$? On $S^2$? On the hyperbolic plane?
\item Derive the Euler-Lagrange equations.
\item Prove that a closed surface in $\R^3$ cannot have everywhere negative gaussian curvature.
\item Let $M$ contain a totally geodesic surface $S$. What can you say about the curvature of $S$?
\item Give an example of a vector field in $\R^2$ which is not uniquely integrable.
\item A Riemannian metric naturally identifies $TX$ and $T^*X$. What kind of form does a volume-preserving flow correspond to?
\item 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?
\item What is the condition on a $1$-form $\alpha$ on a three manifold that $\ker(\alpha)$ be integrable?
\item Is the space of Riemannian metrics on $M$ path connected, for a fixed smooth structure? Is it contractible for $M=S^1$? What about $M=S^2$?
\item What is the exponential map for a Riemannian manifold? [\emph{Pugh}]
\item What is a geodesic? Do they always exist? Are they unique? What are the Christoffel symbols? [\emph{Pugh}]
\item Let $\gamma_v(t)$ denote the geodesic through $p \in M$ with initial tangent vector $v \in T_pM$. Why is $\gamma_{tv}(1) = \gamma_v(t)$? [\emph{Pugh}]
\item What does it mean for two points $p,q \in M$ to be conjugate? [\emph{Weinstein}]
\item 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. [\emph{Weinstein}]
\item Let $S$ be a surface homeomorphic to $S^2$. Let $p,q$ be two points on $S$ and $\gamma$ a minimizing geodesic connecting them. Prove that there must exist at least one more geodesic connecting $p$ and $q$. [\emph{Weinstein}]
\item What is a symplectic manifold? Why must the dimension be even? List all the symplectic manifolds you know. (!) [\emph{Weinstein}]
\item Why is $S^1 \times S^3$ not symplectic? What about $\C P^2\# \C P^2$?
\item Give an example of a symplectic group action which is not hamiltonian.
\item Give an example of a hamiltonian group action which is not Poisson.
\item Given an example of a non-integrable almost-complex structure.
\item Calculate the moment map for the standard action of $SO(3,\R)$ on $\R^3$.
\item Calculate the symplectic volume of $S^{2n+1}_{\sqrt{2E}}$ as a function of $E$, where this denotes the inverse image of $2E$ 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 $\C^{n+1}$. [\emph{Weinstein}]
\item Give a counterexample to Moser’s theorem if $M$ is not compact.
\item Explain geodesics to a symplectic geometer.
\item Let $(M,\omega)$ be symplectic where $\omega$ is actually an integral class. Can you find a principal $S^1$ bundle $P$ over $M$ and a connection form $\theta$ on $P$ such that the curvature of $\theta$ is precisely $\omega$?
\item Let $P$ and $\theta$ be constructed as above. Show that the horizontal distribution on $P$ defined by $\theta$ is a contact structure on $P$.
\item In the setup above, what is $P$ if $M= \C P^n$ with the canonical symplectic structure? Can you calculate $\theta$ in this case?[\emph{Weinstein}]
\item Again, let $P,\theta$ 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 $\theta$ invariant. Is this action hamiltonian (on M)?
\item Characterize the Lagrangian submanifolds of $T^*X$ with the canonical symplectic structure which are “close” to the $0$-section.
\item Let $(M,\omega)$ 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 $T_xX$ is coisotropic. What can you say about $X$?
\item Let $(M,\omega)$ and $X$ be as above, but now suppose that $T_xX$ is a symplectic subspace of $T_xM$ for each $x \in X$. What can you say about $X$?
\item Given $(M,\omega)$ symplectic, why is the space of compatible almost-complex structures contractible, and what is this fact good for?
\item What is a contact manifold? Give some examples of contact manifolds. [\emph{Weinstein}]
\item What is a moment map? What is symplectic reduction? Give an example. [\emph{Weinstein}]
\item What is the Duistermaat-Heckman theorem? Give an example of its use. [\emph{Borcherds}]
\item What does a Hamiltonian vector field look like in local coordinates? [\emph{Halpern}]
\item What are the geodesics on a Lie group with biinvariant metric? [\emph{Weinstein}]
\item Does the Lie group $SU(2)$ admit a metric with constant curvature? Can you prove this using some transitive isometric group action on it? [\emph{Weinstein}]
\item 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? [\emph{Knutsen}]
\end{enumerate}
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