Differential Geometry I
Examination questions will be composed from those below and from questions in
the textbook and previous exams
1
Mappings of manifolds, tensors on manifolds
Algebraic operations on tensors, tensors of type (0, k)
1. Prove that an n-dimensional sphere S n realised as a surface in Rn+1 : x21 +· · ·+x2n+1 = 1, is
a manifold by introducing two subsets UN = S n −(0, . . . , 0, 1) and US = S n −(0, . . . , 0, −1),
and considering the stereographic projection onto the plane xn+1 = 0.
2. Prove that the special unitary group SU (2) of 2 by 2 matrices satisfying U † U = I2 ,
det U = 1 is a 3-dimensional sphere S 3 .
3. (Use a computer for this question). Let h: R3 → R4 be a map defined by
z y
z
y x
y y
x
h : (x1 , x2 , x3 ) = (x, y, z) → (y 1 , y 2 , y 3 , y 4 ) = ( √ + , √ − , + √ , − √ )
2 2 2 2 2
2 2
2
(a) Show that h induces a map f of S 2 ∈ R3 on S 3 ∈ R4 where S 2 and S 3 are given by
the equations x2 + y 2 + z 2 = 1 and (y 1 )2 + (y 2 )2 + (y 3 )2 + (y 4 )2 = 1, respectively.
(b) Introduce local coordinates on S 2 and S 3 using the stereographic projection from
Question 1, and calculate the push-forward f∗ XN and f∗ XS where N and S denote
the north and south poles, and XN is a vector with components (1, 0) in US while
XS is a vector with components (0, 1) in UN .
2
Operations on tensors, tensors of type (0, k)
4. Prove that the results of applying the operations of
(a) Permutation of indices
(b) Contraction
(c) Product of tensors
to tensors are again tensors.
5. A second-rank tensor is called nonsingular if as a matrix it is nonsingular at every point
of M . Show that for such a tensor, the inverse of the matrix is also a tensor.
i ···i
6. Prove that if if Tj11···jqp is a tensor field on a manifold with coordinates x1 , . . . , xn in some
chart U , then the quantities
i ···i
∂Tj11···jqp
i1 ···ip
Tj1 ···jq ;k =
∂xk
1
transform like the components of a tensor of type (p, q + 1) under all linear coordinate
changes
xi = aij z j ,
z i = bij xj ,
aij bjk = δji ,
aij , bij = const ,
i, j, k = 1, . . . , n .
7. Calculate the dimension of the space of k-forms at a given point.
8. Prove that
X
ajik i1 ···ik−1 jik+1 ···in = i1 ···in ajj
k
9. Let ω j = aji dxi . Establish the formula
···ik
dxj1 ∧ · · · ∧ dxjk ,
ω i1 ∧ · · · ∧ ω ik = Jji11···j
k
···ik
where Jji11···j
is the k × k minor of (aji ) formed by the intersections of the rows numbered
k
i1 , . . . , ik with the columns numbered j1 , . . . , jk . Thus in particular
ω 1 ∧ · · · ∧ ω n = det(aji )dx1 ∧ · · · ∧ dxn .
10. Definition. Given a skew-symmetric tensor T of type (0, k) (or a k-form) with components Ti1 ···ik , we define its Hodge dual to be the skew-symmetric tensor ∗T of type (0, n−k)
by the formula
1p
(∗T )ik+1 ···in =
|g|T i1 ···ik i1 ···in ,
k!
where g = det(gij ), and T i1 ···ik = g i1 j1 · · · g ik jk Tj1 ···jk .
(a) Show that
∗(∗T ) = (−1)k(n−k) sgn(g)T
(b) For
ω1 =
1
Ti ···i dxi1 ∧ · · · ∧ dxip ,
p! 1 p
define
{ω1 , ω2 } =
ω2 =
1
Si ···i dxi1 ∧ · · · ∧ dxip ,
p! 1 p
1 i1 j1
g · · · g ip jp Ti1 ···ip Sj1 ···jp .
p!
Show that
ω1 ∧ ∗ω2 = {ω1 , ω2 }
p
|g| dx1 ∧ · · · ∧ dxn .
11. Let U be a bounded region with smooth boundary in an n-dimensional space with metric
gij , and let ΩpU be the space of all smooth p-forms on that space which vanish outside U .
Define a scalar product on the space ΩpU by
Z
hω1 , ω2 i =
ω1 ∧ ∗ω2
U
for every pair of p-forms ω1 , ω2 in ΩpU .
(a) Show that the space ΩpU with this scalar product is Euclidean.
2
(b) The operator ∗ is orthogonal
h∗ω1 , ∗ω2 i = hω1 , ω2 i .
(c) The operators δ = (−1)np+n+1 ∗ d ∗, and d are “conjugate”, i.e.
hdω1 , ω2 i = hω1 , δω2 i .
(d) The square of the operator δ is zero: δδ = 0.
(e) Write ∆ = dδ + δd. Show that the operator ∆ on the space ΩpU is self-conjugate
h∆ω1 , ω2 i = hω1 , ∆ω2 i .
(f) Verify the following commutativity relations
∆d = d∆ ,
3
∆δ = δ∆ ,
∆∗ = ∗∆ .
Surfaces in Euclidean space, projective spaces
12. Definition 1.1.3. A manifold M is said to be oriented if for every pair Up , Uq of
intersecting local coordinate neighbourhoods, the Jacobian Jpq = det(∂xαp /∂xβq ) of the
transition function is positive.
Definition 2.1.2. A manifold M is said to be orientable if it is possible to choose
at every point of it a single orientation class depending continuously on the points. A
particular choice of such an orientation class for each point is called an orientation of the
manifold, and a manifold equipped with a particular orientation is said to be oriented. If
no orientation exists the manifold is non-orientable.
Proposition 2.1.3. Definition 1.1.3. is equivalent to Definition 2.1.2.
Prove the proposition.
13. Theorem 2.1.6. A two-sided hypersurface in Rn is orientable.
Prove the theorem.
14. Prove that the odd-dimensional projective spaces RP 2k+1 are orientable.
15. Prove that the projective spaces RP n and CP n are compact.
4
Elements of Lie groups
16. Prove that every Lie group is orientable.
17. Let F1 (t) and F2 (t) be two one-parameter subgroups of a Lie group G with Ai = F˙ i (0),
whence Fi (t) = exp Ai t, i = 1, 2. Prove that (g0 is the identity element)
F1 (t)F2 (t)F1−1 (t)F2−1 (t) = g0 + t2 [A1 , A2 ] + O(t3 ) .
3
(4.1)
18. Prove that, as operator, dtd Ad F (t)|t=0 is given by B → [A, B] for B in the Lie algebra
which is identifiable with Rn . We denote the map B → [A, B] by ad A : Rn → Rn .
19. Given a curve in the form g(τ ) = F1 (τ t1 ) · · · Fn (τ tn ), prove that
n
X
dg
|τ =0 =
ti Ai .
dτ
i=1
20. Show that the Euler angles φ, ψ, θ constitute coordinates of the second kind on SO(3).
21. Prove that if ρ is a representation of G; ρ : G → GL(N, R) then the induced or pushforward map ρ∗ at the identity element g0 = id is a Lie algebra homomorphism: it is
linear and preserves commutators ρ∗ [ξ, η] = [ρ∗ ξ, ρ∗ η].
f R) of all transformations of the real line of the form
22. Consider the set SL(2,
x → x + 2πa +
1 1 − ze−ix
ln
,
i
1 − z¯eix
where x ∈ R, a ∈ R, z ∈ C, |z| < 1 and ln is the main branch of the natural logarithmic
function, i.e. the continuous branch determined by ln 1 = 0.
f R) is a connected 3-dimensional Lie group.
(a) Show that SL(2,
(b) Calculate its Lie algebra and show that it is isomorphic to sl(2, R).
5
Complex manifolds
23. Prove that the real Jacobian of the transition function from xαq , yqα to xαp , ypα is
∂z α 2
q J = |J | = det
,
∂zpβ
C 2
R
α
α
where zp,q
= xαp,q + iyp,q
.
24. Let D be a region of the complex space C n with complex coordinates z 1 , . . . , z n . We denote
by DR the “realisation” of D with coordinates x1 , . . . , xn , y 1 , . . . , y n where z k = xk + iy k ,
and introduce the complex basis for tangent vectors
1 ∂
∂
∂
= ( k − i k),
k
∂z
2 ∂x
∂y
∂
1 ∂
∂
= ( k + i k),
k
∂ z¯
2 ∂x
∂y
k = 1, . . . , n .
Any complex vector ξ in the tangent space at a point of D is
ξ = ξk
∂
¯ ∂
+ ξk k ,
k
∂z
∂ z¯
The vector ξ is real (i.e. ξ is a linear combination of
¯
ξ¯k = ξ k .
4
¯
ξ = (ξ k , ξ k ) .
∂
, ∂
∂xk ∂y k
with real coefficients) if
The one-forms
dz k = dxk + idy k ,
form the dual basis to
∂
, ∂ .
∂z k ∂ z¯k
d¯
z k = dxk − idy k
Any complex k-form ω, 1 ≤ k ≤ 2n, may be written as
ω = ωk,0 + ωk−1,1 + · · · + ω0,k ,
where
1
i1
ip
T
z j1 ∧ · · · ∧ d¯
z jq
¯ ¯ dz ∧ · · · ∧ dz ∧ d¯
p!q! i1 ...ip j1 ...jq
is called a form of type (p, q).
ωp,q =
(a) Show that the differential operator d on complex forms can be uniquely written as
d = d0 + d00
or
d = ∂ + ∂¯
¯ p,q has type (p, q + 1). The operators ∂, ∂¯ are
where ∂ωp,q has type (p + 1, q) and ∂ω
invariant with respect to complex analytic coordinate changes.
(b) Show that
∂ 2 = ∂¯2 = 0 ,
¯ .
∂ ∂¯ = −∂∂
¯ = 0.
Definition. A form ω of type (p, 0) is called holomorphic if ∂ω
25. Definition 1. A Hermitian metric on a region D of complex n-space with coordinates
z 1 , . . . , z n is given by a family of functions gj k¯ with the following properties
(i) gj k¯ = gkj ;
0
0
0
(ii) under coordinate changes z j = z j (z 1 , . . . , z n ) which are complex analytic, i.e. ∂z j /∂ z¯k =
0, we have
∂z j ∂z k
gj 0 k¯0 = gj k¯ j 0 k0
∂ z¯ ∂ z¯
j ¯k
(iii) the form gj k¯ ξ ξ is positive definite.
Definition 2. The complex scalar product of a pair of complex vectors (ξ k ), (η k ) is
defined in terms of the Hermitian metric gj k¯ by
¯
hξ, ηiC = gj k¯ ξ j η k =⇒ hξ, ηiC = hη, ξiC .
Definition 3. The Hermitian metric on D induces a Riemannian metric on the realised
region DR by hξ, ηiR = Rehξ, ηiC .
(a) Show that the expression {ξ, η} = − 21 Imhξ, ηiC is real and skew-symmetric, and
yields a real differential form Ω of degree 2 on DR :
i
Ω = gj k¯ dz j ∧ d¯
zk ,
2
which is of type (1, 1).
(b) Definition 4. A Hermitian metric gj k¯ on a region D of Cn is said to be K¨ahlerian
if the form Ω = 2i gj k¯ dz j ∧ d¯
z k , is closed, i.e. dΩ = 0.
Show that the Hermitian metric of the form gj k¯ = ∂z∂ j ∂∂z¯k K is K¨ahlerian. The scalar
K is called the K¨ahler potential.
5
6
Homogeneous spaces
26. Prove that O(n) acts on the Stiefel manifold Vn,k transitively.
27. Prove that the Stiefel manifold Vn,k is a nonsingular surface of dimension nk − k(k + 1)/2
in Rnk .
28. Let M be a homogeneous space of G, and let H be the isotropy group. Prove that the
dimension of M is dim M = dim G − dim H. Compute dim Gn,k .
7
The induced metric and pull-back
29. Let an n-dimensional sphere S n of radius R be realised as a surface
n+1
X
x2i = R2 ,
i=1
in the Euclidean space Rn+1 with the standard metric
2
ds =
n+1
X
dx2i .
i=1
It is the boundary the n + 1-dimensional ball (disc) Dn+1 :
Pn+1
i=1
x2i ≤ R2 .
Introduce the spherical coordinates in Rn+1
x1 = r cos θ1 ,
x2 = r sin θ1 cos θ2 ,
x3 = r sin θ1 sin θ2 cos θ3 ,
·
·
·
xn = r sin θ1 sin θ2 · · · sin θn−1 cos θn ,
xn+1 = r sin θ1 sin θ2 · · · sin θn−1 sin θn .
(a) What is the range of r, and θk , k = 1, . . . , n?
(b) Find the metric on Rn+1 and the volume form of Rn+1 in terms of the spherical
coordinates.
(c) The metric on the n-sphere is the metric induced from the ambient Euclidean metric.
Find the metric on the n-sphere and the volume form of S n in terms of the spherical
coordinates. Note that the metric of the unit n-sphere is often denoted by dΩ2n .
(d) Compute the volumes of S n and Dn+1 .
6
30. A hyperbolic n-space, H n , is one sheet of the hyperboloid of two sheets realised as a
surface
n
X
−x20 +
x2i = −R2 , x0 > 0 ,
i=1
1,n
in the Minkowski space R
with the standard metric
2
ds =
−dx20
+
n
X
dx2i .
i=1
Introduce the hyperbolic coordinates in R1,n
x0 = r cosh ρ ,
xi = r yi sinh ρ ,
n
X
i = 1, . . . , n ,
yi2 = 1 ,
i=1
and the spherical coordinates on the unit n-1-sphere S n−1 defined by the coordinates yi .
(a) What is the range of r and ρ?
(b) Find the metric on R1,n and the volume form of R1,n in terms of the hyperbolic
coordinates, and express them through the metric dΩ2n−1 and the volume form ΩS n−1
of the unit S n−1 .
(c) The metric on the hyperbolic n-space is the metric induced from the ambient Minkowski
metric. Find the metric on the hyperbolic n-space and the volume form of H n in
terms of the hyperbolic coordinates, and express them through the metric dΩ2n−1 and
the volume form ΩS n−1 of the unit S n−1 . Note that the metric of the unit hyperbolic
n-space (R = 1) is often denoted by dHn2 . The coordinates ρ and angles θi of the
S n−1 are often referred to as the global coordinates of H n .
31. An n-dimensional de Sitter space, dSn is the hyperboloid of one sheet realised as a surface
−x20 +
n
X
x2i = R2 ,
i=1
in the Minkowski space R1,n with the standard metric
2
ds =
−dx20
+
n
X
dx2i .
i=1
Introduce the hyperbolic coordinates in R1,n
x0 = ρ sinh t ,
xi = ρ yi cosh t ,
n
X
i = 1, . . . , n ,
yi2 = 1 ,
i=1
and the spherical coordinates on the unit n-1-sphere S n−1 defined by the coordinates yi .
7
(a) What is the range of ρ and t?
(b) Find the metric on R1,n and the volume form of R1,n in terms of the hyperbolic
coordinates, and express them through the metric dΩ2n−1 and the volume form ΩS n−1
of the unit S n−1 .
(c) What is the de Sitter space topologically?
(d) The metric on the de Sitter space is the metric induced from the ambient Minkowski
metric. Find the metric on the de Sitter space and the volume form of dSn in terms
of the hyperbolic coordinates, and express them through the metric dΩ2n−1 and the
volume form ΩS n−1 of the unit S n−1 . The coordinates t and angles θi of the S n−1 are
often referred to as the closed slicing coordinates or global coordinates of dSn .
32. An n-dimensional de Sitter space, dSn is the hyperboloid of one sheet realised as a surface
−x20
+
n
X
x2i = R2 ,
i=1
in the Minkowski space R1,n with the standard metric
2
ds =
−dx20
+
n
X
dx2i .
i=1
Introduce the static coordinates for dSn as follows
√
x0 = R 1 − r2 sinh t ,
√
x1 = R 1 − r2 cosh t ,
xi+1 = R r zi , i = 1, . . . , n − 1 ,
n−1
X
zi2 = 1 ,
i=1
and the spherical coordinates on the unit n-2-sphere S n−2 defined by the coordinates zi .
(a) What is the range of r and t?
(b) The metric on the de Sitter space is the metric induced from the ambient Minkowski
metric. Find the metric on the de Sitter space and the volume form of dSn in terms
of the hyperbolic coordinates, and express them through the metric dΩ2n−2 and the
volume form ΩS n−2 of the unit S n−2 .
33. An n-dimensional anti-de Sitter space, AdSn is the hyperboloid of one sheet realised as a
surface
n
X
2
2
x2i = −R2 ,
−x−1 − x0 +
i=1
in the pseudo-Euclidean space R
2
2,n−1
ds =
with the standard metric
−dx2−1
−
dx20
+
n−1
X
i=1
8
dx2i .
Introduce the following coordinates in AdSn
x−1
x0
xi
n−1
X
−y02 +
yi2
= Ry0 cos t , y0 > 0 , −π < t ≤ π ,
= Ry0 sin t ,
= Ryi , i = 1, . . . , n − 1 ,
= −1 ,
i=1
and the hyperbolic coordinates on the hyperbolic n-1-space H n−1 defined by the coordinates yi .
(a) The metric on the anti-de Sitter space is the metric induced from the ambient pseudoEuclidean metric. Find the metric on the anti-de Sitter space and the volume form
of AdSn in terms of the hyperbolic coordinates, and express them through the metric
2
dHn−1
and the volume form ΩH n−1 of the unit H n−1 . The coordinates t and ρ, θi of
n−1
the H
are often referred to as the global coordinates of AdSn .
(b) Note that there are closed time-like curves in AdSn , and by this reason, in physics
g n of the hythe anti-de Sitter space is defined as the universal covering space AdS
perboloid, i.e. one takes t ∈ R. What is the universal covering space topologically?
34. Show that if ω1 , ω2 are differential forms, and f is a smooth map from M to N , then
f ∗ (ω1 ∧ ω2 ) = f ∗ (ω1 ) ∧ f ∗ (ω2 ) .
8
The pull-back, vector fields, the Lie derivative
35. Calculate the operator exp[(ax + b)d/dx] .
36. Definition. Given a k-form
ω=
1
Ti ···i dxi1 ∧ · · · ∧ dxik ,
k! 1 k
we define the exterior derivative or differential of this form to be the k + 1-form given by
k+1
X
∂Ti1 ···iq−1 iq+1 ···ik+1 i1
1 ∂Ti2 ···ik+1 i1
1
ik+1
dω =
dx
∧·
·
·∧dx
=
(−1)q−1
dx ∧· · ·∧dxik+1 ,
i
1
k! ∂x
(k + 1)! q=1
∂xiq
(a) Prove that dω is a k + 1-form.
(b) Prove that d(dω) = 0.
(c) Let ω1 , ω2 be differential forms of degree p and q respectively. Prove that
d(ω1 ∧ ω2 ) = dω1 ∧ ω2 + (−1)p ω1 ∧ dω2 .
9
(d) Prove Cartan’s formula. Let ω be a differential form of rank k, and let X1 , . . . , Xk+1
be smooth vector fields. Then the value of the form dω on the fields X1 , . . . , Xk+1 is
given by the following formula
X
ˇ i , . . . , Xk+1 )
(k + 1)dω(X1 , . . . , Xk+1 ) =
(−1)i ∂Xi ω(X1 , . . . , X
i
+
X
ˇ i , . . . , Xk+1 ) .
(−1)i+j ω([Xi , Xj ], X1 , . . . , X
i<j
Here the “check” over a symbol indicates that that symbol is omitted.
37. Prove Leibniz’ formula for the Lie derivative
Lξ (T ⊗ R) = Lξ (T ) ⊗ R + T ⊗ Lξ R ,
where T, R are arbitrary tensors.
38. Let ω1 , ω2 be differential forms. Show that
Lξ (ω1 ∧ ω2 ) = Lξ (ω1 ) ∧ ω2 + ω1 ∧ Lξ (ω2 ) .
39. Let f be a diffeomorphism from a chart U ∈ M to a chart V ∈ N , let ξ1 , ξ2 be vector
fields defined on M , and let ηi = f∗ ξi be the corresponding fields on N . Prove that
f∗ [ξ1 , ξ2 ] = [η1 , η2 ] .
40. Prove that the space of all vector fields on M is a Lie algebra with respect to the bracket
α
∂ξ α
operation given by the commutator: [ξ, η]α = ξ β ∂η
− η β ∂x
β .
∂xβ
41. Let M be a submanifold of N . Show that if vector fields ξ and η are both tangent to
M , then their commutator is also tangent to the surface. Thus, the linear space of vector
fields tangent to a submanifold is a subalgebra of the Lie algebra of all vector fields.
42. Linear Vector Fields. Let X = (xik ) be a fixed n × n matrix. For each X we construct
a vector field TX on Rn by taking its value at each point x ∈ Rn to be the negative of the
result of applying the matrix X to the vector x
TX (x) = −Xx
⇐⇒
TXi (x) = −Xki xk .
(8.2)
We call such a field TX a linear vector field. Prove that the integral curve x(t) of the vector
field (8.2), TX , satisfying the initial condition x(0) = x0 , is given by x(t) = exp(−tX)x0 .
43. Let


0 0 0
X =  0 0 −1  ,
0 1 0


0 0 1
Y = 0 0 0 ,
−1 0 0


0 −1 0
Z= 1 0 0 .
0 0 0
(a) Show that [X, Y ] = Z, [Y, Z] = X, [Z, X] = Y , i.e. they form a basis of so(3)
algebra.
10
(b) Consider the linear vector fields associated to X, Y, Z and denote them as LX , LY , LZ
LX = (0, z, −y) ,
LY = (−z, 0, x) ,
LZ = (y, −x, 0) .
Show that the three one-parameter groups corresponding to LX , LY , LZ are the
groups of rotations of R3 about the x-, y-, z-axes respectively.
44. Prove that the commutator of two linear vector fields TX and TY has the form
[TX , TY ] = T[X,Y ] .
Thus, a Lie algebra of n × n matrices is isomorphic to the Lie algebra of linear vector
fields defined by those matrices. In the standard basis the vector fields are identified with
operators
∂
∂TX = −Xki xk i ,
∂x
e.g.
∂
∂
∂
∂
∂
∂
− y , LY = x − z
, LZ = y
−x .
LX = z
∂y
∂z
∂z
∂x
∂x
∂y
Check that
[LX , LY ] = LZ ,
[LY , LZ ] = LX ,
[LZ , LX ] = LY .
45. Left-Invariant Fields. Let X = (xik ) be a fixed n × n matrix. To each X there
2
corresponds the linear transformation A → AX of the space Rn of n × n real matrices
2
A. We denote by LX the linear vector field on Rn which at the point A takes the value
LX (A) = AX .
(a) Show that the integral curve of LX satisfying the initial condition A(0) = A0 is given
by
A = A0 exp(tX) .
(b) Prove that
[LX , LY ] = L[X,Y ] .
2
(c) Let G be a matrix group considered as a smooth surface in Rn of all n × n real
matrices A, and let G be the tangent space to G at the identity. Recall that if X ∈ G
then the matrices exp(tX) form a one-parameter subgroup of G. Show that for each
X ∈ G, the vector field LX is tangent to the surface G; hence its restriction to G is
a vector field on G.
(d) Definition. A vector field of the form LX on a classical group G where X is an
element of the Lie algebra G of G is called a left-invariant field on the group G.
Prove that the left-invariant vector fields on a group G form a Lie algebra isomorphic
to the Lie algebra G of G.
11
46. Invariant Metrics on a Transformation Group
Definition. A Euclidean or pseudo-Euclidean scalar product h , i0 on a Lie algebra G is
called an invariant scalar product if all the operators ad X, X ∈ G, are skew-symmetric
with respect to h , i0 , i.e. if
had X(Y ), Zi0 = −hY, ad X(Z)i0 .
Let G be a Lie algebra of a transformation group G, and suppose an invariant scalar
product h , i0 is given on G. Let us use this scalar product and the left-invariant fields
on G to introduce a metric on the surface G itself. Let A be any point (i.e. matrix) in
G; then every vector tangent to G at A has the form LX (A) for some unique X ∈ G. It
follows that in setting
hLX (A), LY (A)i0 = hX, Y i0
for all X, Y ∈ G we defined the scalar product of any pair of vectors tangent to G at
the arbitrary point A, i.e. we defined a metric on G. This metric is called the invariant
metric on the group G.
Choosing X = Y = A−1 dA where dA = (dAij ) is the matrix of differentials of Aij one gets
the square of the element of length on G
dl2 = hA−1 dA, A−1 dAi0 .
2
(a) Show that the Euclidean metric on the space Rn of all n × n real matrices induces
an invariant metric on SO(n, R).
(b) Let gij0 be an invariant scalar product on a Lie algebra G with basis X1 , . . . , Xn .
Write
0 l
[Xi , Xj ] = ckij Xk , ckij = gkl
cij .
Show that the tensor ckij is skew-symmetric.
(c) An invariant metric on the group SO(p, q) can be obtained as the restriction to this
group of the pseudo-Euclidean metric
hX, Y i = tr(GXGY T ) ,
where G = diag (Ip , −Iq ) is the matrix of the pseudo-Euclidean metric of type (p, q).
What is the type of the resulting pseudo-Riemannian metric on SO(p, q)?
47. Define a right-invariant field on a matrix group G to be the restriction to G of a vector
field of the form RX (A) = −XA. Prove that
[RX , RY ] = R[X,Y ] ,
[LX , RY ] = 0 .
48. Let gab be a metric of Euclidean or pseudo-Euclidean space, i.e. gab = λa δab , λa = ±1.
Consider the vector fields
i) (pseudo-)rotations:
Ωab = gac xc
∂
∂
− gbc xc a ,
b
∂x
∂x
12
a, b = 1, . . . , n
ii) translations:
Pa =
∂
∂xa
iii) dilations or dilatations:
D = xa
∂
∂xa
iv) inversions:
Ka = 2gac xc xb
∂
∂
− gbc xb xc a
b
∂x
∂x
(a) Show that the operator exp(tΩab ) defines either a rotation (if λa = λb ) or a Lorentz
transformation (if λa = −λb ) of the (xa , xb )-plane.
(b) Show that the operator exp(tPa ) defines translations along the xa -axis.
(c) Show that the operator exp(tD) defines dilations
exp(tD)f (x1 , . . . , xn ) = f (et x1 , . . . , et xn ).
(d) Show that the vector fields form a Lie algebra which is called the Lie algebra of the
group of conformal transformations of the pseudo-Euclidean space Rp,q , p + q = n
[Ωab , Ωcd ] = gac Ωbd − gbc Ωad + gad Ωcb − gbd Ωca ,
[Ωab , Pc ] = gac Pb − gbc Pa , [Ωab , Kc ] = gac Kb − gbc Ka ,
[Ωab , D] = [Pa , Pb ] = [Ka , Kb ] = 0 ,
[Pa , Kb ] = 2(gab D + Ωab ) , [Pa , D] = Pa , [Ka , D] = −Ka .
(e) Show that the Lie algebra of the group of conformal transformations of the pseudoEuclidean space Rp,q , p + q = n is isomorphic to the Lie algebra so(p + 1, q + 1) by
identifying the generators as
Ωab ←→ Jab , D ←→ Jn+1,n+2 , a, b = 1, . . . , n ,
Pa ←→ Ja,n+1 − Ja,n+2 , Ka ←→ Ja,n+1 + Ja,n+2 .
49. Conformal transformations of Euclidean and pseudo-Euclidean spaces
0
Definition 1. The two metrics gab
(x) and gab (x) on a manifold M are said to define the
0
same conformal structure on M , or are conformally equivalent, if gab
(x) = λ(x)gab (x) for
some real-valued function λ(x) > 0.
0
Thus, a metric gab
(y) given to us in terms of coordinates y 1 , . . . , y n is conformally equivalent to gab (x) if after expressing the two metrics in the same coordinates they differ only
by a factor.
Definition 2. A diffeomorphism between two Riemannian manifolds is called a conformal
map if the pulled-back metric is conformally equivalent to the original one.
(a) Show that the stereographic projection of a sphere onto a plane augmented with a
point at infinity is a conformal map.
13
(b) Let M = Rp,q , p + q = n. Compute the Lie derivatives (strain tensors) of the
metric gab = λa δab , λa = ±1 along the vector fields Ωab , Pa , Ka , D and show that
they are proportional to gab . Thus, any transformation of the form exp(tA) where
A = λab Ωab + µa Pa + δ a Ka + γD is (locally) a conformal transformation of Rp,q for
at least small enough t.
(c) Show that the inversion
xa →
xa − xa0
,
hx − x0 , x − x0 i
hx − x0 , x − x0 i = gab (x)(xa − xa0 )(xb − xb0 )
is a conformal transformation of Rn (one also adds a point at infinity to Rn to define
the map at x = x0 ).
50. Let X1 , . . . , Xn be vector fields defines on an n-dimensional manifold M , linearly independent at each point of M , and let ω 1 , . . . , ω n be at each point the dual basis of 1-forms,
i.e. ω i (Xj ) = δji . Show that
1
dω k = − ckij ω i ∧ ω j ,
2
k
where the quantities cij are defined by [Xi , Xj ] = ckij Xk .
51. Show that the operations of taking the differential and taking the Lie derivative of forms
commute, i.e.
Lξ dω = d(Lξ ω) .
52. For each vector field X define a linear operator i(X) on forms by
[i(X)ω](X1 , . . . , Xk−1 ) = ω(X, X1 , . . . , Xk−1 ) ,
where ω is any form of rank k, and X1 , . . . , Xk−1 are arbitrary vector fields.
(a) Prove that i(X) is “anti-differentiation”, i.e.
i(X)(ω1 ∧ ω2 ) = (i(X)ω1 ) ∧ ω2 + (−1)k ω1 ∧ i(X)ω2 ,
where k is the rank of ω1 .
(b) Establish the formula
i(X) d + d i(X) = LX ,
where LX denotes the operation of taking the Lie derivative along the field X.
9
Covariant differentiation, parallel transport
53. Show that if a vector (ξ i ) is parallely transported through a small interval (δxk ) then its
components change as follows
ξ i → ξ i − ξ j Γijk δxk + O(|δx|2 ) .
54. Express the strain tensor in terms of covariant derivatives.
14
55. The equation of motion of a point electric charge in the field of a magnetic pole has the
form
r × r˙
¨r = a 3 , a = const .
|r|
Prove that the trajectory of a point charge is a geodesic on a circular cone.
56. Find all the geodesics of the Lobachevsky plane with the metric
ds2 =
4
(1 −
x2
− y 2 )2
(dx2 + dy 2 ) ,
x2 + y 2 < 1 .
57. Show that the geodesics of the sphere are just the great circles.
58. Prove that the level curves of the function
Z
Z
du
dv
p
z(u, v) =
± p
f (u) − a
g(v) + a
are geodesics with respect to the metric
ds2 = (f (u) + g(v))(du2 + dv 2 ) ,
f > 0, g > 0.
59. Given that Γijk is a symmetric connection compatible with the metric, show that
(a)
1 ∂ p
( |g|g ik ) .
g kl Γikl = − p
k
∂x
|g|
(b)
Γiki =
1 ∂g
.
2g ∂xk
60. Given a metric in the form
ds2 = grr dr2 + r2 dϕ2 ,
show that the line ϕ = ϕ0 through the point r = 0 (the origin) is geodesic.
61. Let M be a surface in Euclidean n-space Rn , let π be the linear operator which projects
Rn orthogonally onto the tangent space to M at an arbitrary fixed point of M , and let
X, Y be vector fields in Rn tangent to the surface M . Show that the connection on M
compatible with the induced metric on M satisfies
∇X Y = π(X k
∂Y
).
∂xk
62. Use any method and any coordinates you like to calculate the components of the LeviCivita connection of
(a) an n-sphere S n ,
(b) a hyperbolic n-space H n ,
15
(c) a de Sitter n-space dSn ,
(d) an anti-de Sitter n-space AdSn .
63. The line-element on the unit 2-sphere in spherical coordinates is given by
ds2 = dθ2 + sin2 θdφ2 .
(a) Calculate the components of the Levi-Civita connection.
(b) A covector field on this 2-sphere with components Am in the coordinate system
(θ, φ) is taken with initial components (X, Y ) and is carried by parallel transport
with respect to the metric connection along an arc of the circle θ = α = const of
length φ sin α. Show that the components Am attain finsl values
Aθ = X cos(φ cos α) + Y csc α sin(φ cos α) ,
Aφ = −X sin α sin(φ cos α) + Y cos(φ cos α) .
64. Consider the 2-dimensional de Sitter space with the metric
ds2 = −dt2 + cosh2 tdθ2 .
(a) Calculate the components of the metric connection.
(b) Calculate the components of the metric curvature tensor.
(c) Show that the geodesics xm (τ ) satisfy
θ˙ cosh2 t = K ,
t˙2 cosh2 t = K 2 − L cosh2 t ,
where a dot ˙ denotes differentiation with respect to τ , andK and L are constants
that should be identified.
(d) The first equation above allows one to choose θ as a parameter of the geodesics.
Then the second equation takes the form
t02
K2
= K 2 − L cosh2 t ,
cosh2 t
where 0 denotes differentiation with respect to θ. Find the change of the variable t:
t = f (v) which brings the equation to the form
v 02 = M 2 − v 2 ,
where the constant M should be identified.
(e) Determine and sketch the geodesics (timelike, null and spacelike) passing through
(t, θ) = (0, 0).
16
10
The curvature tensor
˜ be the result of parallel-transporting a vector ξ = (ξ k ) around the boundary of
65. Let ξ()
a square of side with its sides parallel to the coordinates xi - and xj -axes. Prove that
ξ˜k () − ξ k
k l
ξ .
= −Rlij
→0
2
lim
66. Prove Bianchi’s identities for the curvature tensor of a symmetric connection
n
n
n
∇m Rikl
+ ∇l Rimk
+ ∇k Rilm
= 0.
67. Prove the formula for the divergence of the Ricci tensor of the symmetric connection
compatible with a given metric
1 ∂R
∇m Rim =
.
2 ∂xi
68. Show that in 2D, the curvature tensor is given by
1
Rabcd = R(gac gbd − gad gbc ) ,
2
and evaluate the Einstein tensor in two dimensions.
69. Show that if dim (M ) ≥ 3 and
1
Rabcd = K(gac gbd − gad gbc ) ,
2
where K is a scalar function, then K must be a constant.
70. Prove that in 3D, the curvature tensor is given by
1
Rabcd = Rac gbd − Rad gbc + gac Rbd − gad Rbc − R(gac gbd − gad gbc ) .
2
71. Show that the 2D Riemannian manifold with line-element
ds2 = e2φ (dx2 + dy 2 )
has constant scalar curvature R if and only if φ satisfies the Liouville equation
∂ 2 φ ∂ 2 φ R 2φ
+
+ e = 0.
∂x2 ∂y 2
2
72. Consider the 2D manifold with line-element
ds2 = du2 + 2 cos φ dudv + dv 2 ,
where φ can be either real or pure imaginary.
(a) Determine when the manifold is Riemannian and when it is pseudo-Riemannian.
17
(b) Show that it has constant scalar curvature R if and only if φ satisfies the sine-Gordon
equation
R
∂ 2φ
+ sin φ = 0 ,
∂u∂v
2
which takes the form of the sinh-Gordon equation if φ = iϕ is imaginary
∂ 2ϕ
R
+ sinh ϕ = 0 ,
∂u∂v
2
(c) Find a change of the coordinates u, v to new coordinates t, x in terms of which the
sine-Gordon equation takes the form
∂ 2φ ∂ 2φ R
− 2 + sin φ = 0 .
∂t2
∂x
2
73. Use any method and any coordinates you like to calculate the components of the Riemann
tensor, Ricci tensor, and scalar curvature of
(a) an n-sphere S n ,
(b) a hyperbolic n-space H n ,
(c) a de Sitter n-space dSn ,
(d) an anti-de Sitter n-space AdSn .
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