\\title{Sample Questions from Past Qualifying Exams}
\nThis 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.<\/p>\n
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).<\/p>\n
The listing by section is approximate, since some questions may fit under more than one heading.
\n
\n\\section{Geometry, Riemannian and Symplectic}
\n\\begin{enumerate}
\n\\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 $q$?
\n\\item List all the Jacobi fields along a great circle joining the north and south poles on $S^n$[\\emph{Weinstein}]
\n\\item Find two equal-volume flat tori that are not isometric.
\n\\item Compute the curvature of the unit sphere in $\\R^3$.
\n\\item What is parallel translation? How is it related to the notion of a connection on a principal bundle?
\n\\item Given a principal $O(n)$ bundle, give some examples of associated vector bundles.
\n\\item Give $S^2$ the usual induced metric from $\\R^3$ and the (local) parameterization
\n \\[
\n (\\theta,\\phi) \\to (\\cos \\phi \\cos \\theta, \\cos \\phi \\sin \\theta,
\n \\sin \\phi)
\n \\]
\nConstruct 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.
\n\\item What are the geodesics on $\\R^n$? On $S^2$? On the hyperbolic plane?
\n\\item Derive the Euler-Lagrange equations.
\n\\item Prove that a closed surface in $\\R^3$ cannot have everywhere negative gaussian curvature.
\n\\item Let $M$ contain a totally geodesic surface $S$. What can you say about the curvature of $S$?
\n\\item Give an example of a vector field in $\\R^2$ which is not uniquely integrable.
\n\\item A Riemannian metric naturally identifies $TX$ and $T^*X$. What kind of form does a volume-preserving flow correspond to?
\n\\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?
\n\\item What is the condition on a $1$-form $\\alpha$ on a three manifold that $\\ker(\\alpha)$ be integrable?
\n\\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 $M=S^2$?
\n\\item What is the exponential map for a Riemannian manifold? [\\emph{Pugh}]
\n\\item What is a geodesic? Do they always exist? Are they unique? What are the Christoffel symbols? [\\emph{Pugh}]
\n\\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}]
\n\\item What does it mean for two points $p,q \\in M$ to be conjugate? [\\emph{Weinstein}]
\n\\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}]
\n\\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}]
\n\\item What is a symplectic manifold? Why must the dimension be even? List all the symplectic manifolds you know. (!) [\\emph{Weinstein}]
\n\\item Why is $S^1 \\times S^3$ not symplectic? What about $\\C P^2\\# \\C P^2$?
\n\\item Give an example of a symplectic group action which is not hamiltonian.
\n\\item Give an example of a hamiltonian group action which is not Poisson.
\n\\item Given an example of a non-integrable almost-complex structure.
\n\\item Calculate the moment map for the standard action of $SO(3,\\R)$ on $\\R^3$.
\n\\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
\n \\[
\n \\mu:(z_0, \\dots, z_n)) \\to 1\/2(|z_0|^2 + \\dots + |z_n|^2)
\n \\]
\n for the standard $S^1$ action on $\\C^{n+1}$. [\\emph{Weinstein}]
\n\\item Give a counterexample to Moser’s theorem if $M$ is not compact.
\n\\item Explain geodesics to a symplectic geometer.
\n\\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 $\\omega$?
\n\\item Let $P$ and $\\theta$ be constructed as above. Show that the horizontal distribution on $P$ defined by $\\theta$ is a contact structure on $P$.
\n\\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}]
\n\\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)?
\n\\item Characterize the Lagrangian submanifolds of $T^*X$ with the canonical symplectic structure which are “close” to the $0$-section.
\n\\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 $X$?
\n\\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 $X$?
\n\\item Given $(M,\\omega)$ symplectic, why is the space of compatible almost-complex structures contractible, and what is this fact good for?
\n\\item What is a contact manifold? Give some examples of contact manifolds. [\\emph{Weinstein}]
\n\\item What is a moment map? What is symplectic reduction? Give an example. [\\emph{Weinstein}]
\n\\item What is the Duistermaat-Heckman theorem? Give an example of its use. [\\emph{Borcherds}]
\n\\item What does a Hamiltonian vector field look like in local coordinates? [\\emph{Halpern}]
\n\\item What are the geodesics on a Lie group with biinvariant metric? [\\emph{Weinstein}]
\n\\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}]
\n\\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}]
\n\\end{enumerate}<\/p>\n","protected":false},"excerpt":{"rendered":"
\\title{Sample Questions from Past Qualif …<\/p>\n