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This is an exam paper for the differential geometry course in the mathematical tripos part iii program at the university of cambridge. It covers topics such as smooth vector fields, flows, lie algebras, connections on vector bundles, and riemannian metrics. The exam consists of five questions, and students are required to attempt no more than four. Each question carries equal weight.
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Tuesday, 5 June, 2012 1:30 pm to 4:30 pm
Attempt no more than FOUR questions. There are FIVE questions in total. The questions carry equal weight.
Cover sheet None Treasury Tag Script paper
You may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator.
Let X be a smooth vector field on a manifold M.
State and prove a uniqueness result for integral curves of X that pass through a given point p ∈ M. Using this, or otherwise, show that if φt is the flow of X then
φt+s = φt ◦ φs
whenever both sides are defined. [Standard results from the theory of Ordinary Differential Equations may be assumed without proof.]
Now suppose that Y is another smooth vector field with flow ψs and that the flows of X and Y commute (i.e. φt ◦ ψs = ψs ◦ φt whenever both sides are defined). Show that
(Dqφt)(Yq ) = Yφt(q) for all q ∈ M. (1)
Finally suppose that in addition to the flows commuting, the vector fields X and Y are pointwise linearly independent. Prove that given any point p ∈ M there exists a two dimensional submanifold S ⊂ M containing p such that if γ is any curve in M with γ(0) = p and so ˙γ(t) lies in the plane spanned by Xγ(t) and Yγ(t) for all t, then γ(t) ∈ S for all t [Results from lectures may be assumed if stated clearly.]
Part III, Paper 17
Define what is meant by a connection on a vector bundle E. Prove that E admits a connection, and that the space of connections is (non-canonically) isomorphic to Ω^1 (End(E)).
Now suppose that ∇ is a linear connection on a manifold M (i.e. a connection on T M ). Define a map τ : Vect(M ) × Vect(M ) → Vect(M ) by
τ (X, Y ) = ∇X Y − ∇Y X − [X, Y ].
Show that τ is tensorial (i.e. it is induced by a certain tensor on M whose type you should determine explicitly).
Let {e 1 ,... , en} be a local frame for T M over some open set U. Show that there exist unique 1-forms ωij on U such that
∇X ei =
j
ωij (X)ej
for all i and all X ∈ Vect(M ).
Finally let φi be the frame for T ∗M over U that is dual to ei (i.e. φi(ej ) = δij ). Prove that dφj =
i
φi ∧ ωij + τj
where τ (X, Y ) =
j
τj (X, Y )ej.
Part III, Paper 17
g(Vσ, Y ) = Y (σ) for all Y ∈ Vect(M ).
Show also that if ˜g(X, Y ) = e^2 σg(X, Y ) then ˜g is a well-defined Riemannian metric on M.
∇˜X Y = ∇X Y + X(σ)Y + Y (σ)X − g(X, Y )Vσ
where ˜g is as in the first part of the question. Hence or otherwise prove that if M is compact then there exists a point p ∈ M such that ∇˜X Y |p = ∇X Y |p for all X and Y.
Part III, Paper 17
