MATH 436 HOMEWORK ASSIGNMENT 5. DUE NOV. 30TH
Problem 1: For each of the two manifolds N = S2 and N = H2, choose a point
p ∈ N and compute the pull-back exp∗(µ) where µ : TN⊕TN → R is the standard
Riemann metric on S2 and H2 respectively, and exp : TpN → N (i.e. we restrict
the exponential map to an individual tangent space). In both cases, determine the
injectivity radius,
sup{r ∈ R : exp| : Br(p)→ N is an isometry to its image }
where Br(p) = {v ∈ TpN : |v| < r}. Here ‘compute’ means write out an explicit
formula for exp∗(µ)q(v, w) so that it’s clear exactly which bilinear function this is
for all q ∈ Br(p). The Riemann metric on Br(p) is the pull-back metric. Hint: It
is okay to use polar coordinates in the tangent space.
Problem 2: Let H = {(x, y) ∈ R2 : x > 0}. Give H the Riemann metric (in
‘traditional’ notation)
xdx2 + dy2
In our more explicit notation, we would say
µ((x, y), (v1, v2), (w1, w2)) = xv1w1 + v2w2.
Compute the Christoffel symbols for the Levi-Cevita connection and check that
the x-axis is a geodesic (suitably parametrized). Is H complete with this Riemann
metric?
Problem 3: The flat torus is S1 × S1 ⊂ R4 with the induced Riemann metric
and connection. Prove that for any p, q ∈ R, γ : R → S1 × S1 defined by γ(t) =
(eipt, eiqt) is a geodesic in S1 × S1. Further, argue if p and q are not rational
multiples of each other, that γ is a one-to-one immersion, but not an embedding.
Hint: I think this is most easily handled by thinking of the Ehresmann connection
as being given by orthogonal projection.
Problem 4: Let N ⊂ Rk be the regular value of a smooth function f : U → Rj ,
N = f−1(q) where q ∈ Rj and U ⊂ Rk is open. Then f∗(dx1 ∧ · · · ∧ dxj) is a
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smooth j-form on U . Argue that the Hodge star of this form is a volume form on
N , i.e. nonzero on any tangent space basis.
Problem 5: In a connected, complete Riemann manifold, the exponential map
exp : TpN → N is known to be onto. If this manifold is compact, one can imagine
exp(Br(p)) as the result of blowing up a balloon, centred at p ∈ N . Since the
exponential map is onto, every point in N is in exp(Br(p)) for suitably-large r.
But for a compact manifold, some finite r will do, this is called the diameter of
the manifold. If we imagine inflating an actual balloon, at some point the balloon
will come into contact with itself. Eventually the ‘interior’ of the balloon will
fill the entire manifold. This must give a decomposition of N as a compact ball
with some identifications on the boundary. This is called the Dirichlet Domain
for the manifold. Specifically, define the Dirichlet domain of N , centred at p to
be the subspace of TpN consisting of all v ∈ TpN such that there is a unique
length-minimizing geodesic between p and exp(v), and the geodesic is exp(tv) where
t ∈ [0, 1]. Determine the Dirichlet domain for S1 × S1 ⊂ R4, i.e. the flat torus.
Problem 6: Prove that if M is a 2-dimensional Riemann manifold, and if a
1-dimensional submanifold C ⊂ M is fixed by a non-trivial isometry i : M → M ,
i.e. i 6= IdM yet i(p) = p for all p ∈ C, then C, suitably parametrized, is a geodesic
in M .
Bonus Problem: Can a non-orientable manifold be the pre-image of a regular
value of a smooth function?