Tutorial 11—Global Analysis
1. Suppose is an affine connection on a manifold M.
(a) Show that its curvature, given by,
R(ξ, η)(ζ) = ξ ηζ − η ξζ − [ξ,η]ζ,
for vector fields ξ, η, ζ ∈ X(M) defines a
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3
-tensor on M.
(b) Show that, if is torsion-free, the Bianchi identity holds:
R(ξ, η)(ζ) + R(η, ζ)(ξ) + R(ζ, ξ)(η) = 0,
for any ξ, η, ζ ∈ X(M).
2. Suppose E → M is a vector bundle over a manifold M equipped with a linear
connection , that is, a R-bilinear map
: Γ(TM) × Γ(E) → Γ(E)
(ξ, s) → ξs
such that for ξ ∈ Γ(TM), s ∈ Γ(E) and f ∈ C∞
(M, R) one has
• fξs = f ξs
• ξfs = f ξs + (ξ · f)s.
(a) Show that : Γ(TM)×Γ(E∗
) → Γ(E∗
) (typically also denoted by ) given
by
( ξµ)(s) = ξ · µ(s) − µ( ξs), for µ ∈ Γ(E∗
), ξ, ∈ Γ(TM), s ∈ Γ(E)
defines a linear connection on the dual vector bundle E∗
→ M.
(b) Suppose ˜E → M is another vector bundle equipped with a linear connection
˜ . Show the vector bundle E ⊗ ˜E → M admits a linear connection characterized
by
ξ(s ⊗ ˜s) = ξs ⊗ ˜s + s ⊗ ˜ ξ ˜s
for ξ ∈ Γ(TM), s ∈ Γ(E) and ˜s ∈ Γ( ˜E).
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3. Suppose is an affine connection on a manifold M. Then the previous exercise
shows that induces a linear connection : Γ(TM) × T p
q (M) → T p
q (M) on all
tensor bundles. Show that it also induces a linear connection on the bundles Λk
T∗
M
for k = 1, ... dim(M) characterized by
ξ(ω ∧ µ) = ξω ∧ µ + ω ∧ ξµ
for ω ∈ Γ(Λk
T∗
M) and µ ∈ Γ(Λ T∗
M) and give a formula.
4. Suppose (M, g) is a Riemannian manifold.
(a) For vector fields ξ, η ∈ X(M), let ξη ∈ X(M) be the unique vector field
such that
g( ξη, ζ) =
1
2
ξ·g(η, ζ)+η·g(ζ, ξ)−ζ·g(ξ, η)+g([ξ, η], ζ)−g([ξ, ζ], η)−g([η, ζ], ξ)
for all ζ ∈ X(M). Show that defines a torsion-free affine connection satis-
fying
ξ · g(η, ζ) = g( ξη, ζ) + g(η, ξζ)
for ξ, η, ζ ∈ X(M).
(b) The connection in (a) is called the Levi-Civita connection of (M, g). Show
that its curvature satisfies:
• g(R(ξ, η)(ζ), µ) = −g(R(ξ, η)(µ), ζ),
• g(R(ξ, η)(ζ), µ) = g(R(ζ, µ)(ξ), η),
for ξ, η, ζ, µ ∈ X(M).
(c) Suppose (U, u) is a chart for M and let R be the Riemann curvature, i.e. the
curvature of the Levi-Civita connection of (M, g). Compute
R(
∂
∂ui
,
∂
∂uj
)(
∂
∂uk
)
in terms of the Christoffel symbols.
5. Suppose H = {(x, y) ∈ R2
: y > 0} is the upper-half plane and equip it with the
Riemannian metric
g =
1
y2
dx ⊗ dx +
1
y2
dy ⊗ dy.
(a) Compute the Christoffel symbols of g.
(b) Compute the geodesics of g.
(c) Compute the Riemann curvature.
6. Identifying H = {(x, y) ∈ R2
: y > 0} = {z = x + iy ∈ C : y > 0} in the
previous example, we may write g as
g =
1
Im(z)2
Re(dz ⊗ d¯z) =
4
|z − ¯z|2
Re(dz ⊗ d¯z),
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where dz = dx + idy, d¯z = dx − idy, Im and Re denote imaginary and real part,
and |−| is the absolute value of complex numbers.
Consider SL(2, R) = {A ∈ GL(2, R) : det A = 1}. For A =
a b
c d
∈ SL(2, R)
and z ∈ C let
fA(z) =
az + b
cz + d
.
(a) Show that fA is a diffeomorphism from H to itself for any A ∈ SL(2, R) and
that fAB = fA ◦ fB.
(b) Show that fA is an isometry of H.
(c) Show that for any two points z, z ∈ H there exists A ∈ SL(2, R) such that
fA(z) = z .
(d) Characterize the elements A ∈ SL(2, R) such that fA(i) = i.