Dierential Geometry 1
Homework 02
1. Let M be a smooth manifold. By a smooth curve through p in M we mean
a smooth map : I M from some open interval (0 ) I R such that
(0) = p.
(a) Show that a tangent vector Tp M is dened by the rule
f C (M ) (f ) = (f )

Dierential Geometry 2
Homework 04
1. Let V, (|) be an inner product space and let f be the real-valued function
on V dened by
v V f (v ) = 2/(1 + (v |v ).
Let g0 be the standard Riemannian metric on V and let g = f 2 g0 be the
conformal metric. Explicitly

Dierential Geometry 2
Homework 05
1. Let V and V be lattices in nite-dimensional complex vector
spaces. Show that if F O(V /, V / ) is a holomorphic map between
the corresponding tori then there exist a linear map A L(V, V ) such that
A() and a vector b V

Dierential Geometry 2
Homework 05
1. Let V and V be lattices in nite-dimensional complex vector
spaces. Show that if F O(V /, V / ) is a holomorphic map between
the corresponding tori then there exist a linear map A L(V, V ) such that
A() and a vector b V

Dierential Geometry 2
Homework 06
1. Let M be a connected complex manifold (of complex dimension greater
than one) on which g is a Khler metric. Show that the only Khler metrics
a
a
on M that are conformal to g are the constant multiples of g .
2. Let V,

Dierential Geometry 2
Homework 06
1. Let M be a connected complex manifold (of complex dimension greater
than one) on which g is a Khler metric. Show that the only Khler metrics
a
a
on M that are conformal to g are the constant multiples of g .
2. Let V,

Dierential Geometry
January 2007
Answer SIX questions. Write solutions in a neat and logical fashion,
giving complete reasons for all steps and stating carefully any substantial
theorems used.
1. Explain how each of the following is a smooth manifold:
(a)

Dierential Geometry 2
Test 1
Let (M, ) be a symplectic manifold and let H C (M ) be a smooth function.
Say that F C (M ) is a constant of the H -motion if and only if F is constant
along each integral curve of the Hamiltonian vector eld H .
(a) Prove that

Dierential Geometry 2
Test 2
Dene what is meant by a Lie group G and by its Lie algebra g; dene
also the exponential map exp : g G.
(a) Show that R G : u exp(u ) is the integral curve of g through
the identity e G.
(b) Prove that there exist open sets (0

Dierential Geometry 2
Test 3
Dene what is meant by a Lie group homomorphism : G H and construct its derivative # : g h.
(a) Show that expG = expH # and explain briey why is uniquely
determined by its derivative # when G is connected.
(b) Show that if : g

Dierential Geometry 2
Test 4
(1) Let G be a connected Lie group with Lie algebra g; let K G be a
connected Lie subgroup and k g its Lie algebra. Prove that if k is an ideal
in g then K is normal in G. (The converse is true without connectedness.)
(2) Let

Dierential Geometry 2
Test 5
Let SO(3) be the Lie group of rotations on R3 and let so(3) L(R3 ) be
its Lie algebra.
1. Show that the linear map : R3 R3 lies in so(3) i
x, y R3 x y + x y = 0.
2. Prove that the map
: R3 so(3) : z z ( )
is a Lie algebra iso

Dierential Geometry 1
Homework 01
1. (a) Prove explicitly that the real projective line P 1 R = P (R2 ) is homeomorphic to the unit circle S 1 R2 .
(b) Prove explicitly that the complex projective line P 1 C = P (C2 ) is homeomorphic to the unit sphere S

Dierential Geometry 1
Homework 01
1. (a) Prove explicitly that the real projective line P 1 R = P (R2 ) is homeomorphic to the unit circle S 1 R2 .
(b) Prove explicitly that the complex projective line P 1 C = P (C2 ) is homeomorphic to the unit sphere S

Dierential Geometry 2
Homework 04
1. Let V, (|) be an inner product space and let f be the real-valued function
on V dened by
v V f (v ) = 2/(1 + (v |v ).
Let g0 be the standard Riemannian metric on V and let g = f 2 g0 be the
conformal metric. Explicitly

Dierential Geometry 2
Homework 03
1. Let S V be the unit sphere in a Euclidean space. Recall that if is
V
S
a smooth curve in S then D = (D )T . Use this fact to determine the
(unit-speed) geodesics in S explicitly. Do the same for geodesics in hyperbolic

Dierential Geometry 1
Homework 03
1. Determine the integral curves for the following vector eld on R2 :
1
(y + z )
+ (z + x)
+ (x + y )
.
2
x
y
z
Do so in more than one way!
2. Consider the vector elds and dened on R2 by
= x2
, = y2 .
y
x
Determine which

Dierential Geometry 1
Homework 03
1. Determine the integral curves for the following vector eld on R2 :
1
(y + z )
+ (z + x)
+ (x + y )
.
2
x
y
z
Do so in more than one way!
2. Consider the vector elds and dened on R2 by
= x2
, = y2 .
y
x
Determine which

Dierential Geometry 1
Homework 04
1. Let Vec(S 1 ) be the standard vector eld given by
(a, b) S 1 (a,b) = (a,b) (b, a)
and let 1 (S 1 ) be the standard one-form given by ( ) 1.
(i) Find explicitly the integral curve of through (a, b) S 1 .
(ii) Show that

Dierential Geometry 1
Homework 04
1. Let Vec(S 1 ) be the standard vector eld given by
(a, b) S 1 (a,b) = (a,b) (b, a)
and let 1 (S 1 ) be the standard one-form given by ( ) 1.
(i) Find explicitly the integral curve of through (a, b) S 1 .
(ii) Show that

Dierential Geometry 1
Homework 05
1. Let (x, y, z ) be the standard coordinates on R3 and (X, Y ) those on R2 .
Let 2 (S 2 ) be the standard two-form:
= xdy dz + y dz dx + z dx dy
and let F : R2 S 2 be inverse to stereographic projection from the north
p

Dierential Geometry 1
Homework 05
1. Let (x, y, z ) be the standard coordinates on R3 and (X, Y ) those on R2 .
Let 2 (S 2 ) be the standard two-form:
= xdy dz + y dz dx + z dx dy
and let F : R2 S 2 be inverse to stereographic projection from the north
p

Dierential Geometry 1
Homework 06
Let 1 (M ) be nowhere-zero and choose VecM such that ( ) is
identically one.
1. Show that if k+1 (M ) and = 0 then = for some
k (M ). [Hint: Recall that is an antiderivation.]
2. Assume that d = 0 and choose 1 (M ) such

Dierential Geometry 1
Homework 06
Let 1 (M ) be nowhere-zero and choose VecM such that ( ) is
identically one.
1. Show that if k+1 (M ) and = 0 then = for some
k (M ). [Hint: Recall that is an antiderivation.]
2. Assume that d = 0 and choose 1 (M ) such

Dierential Geometry 2
Homework 01
1. Let V be a nite-dimensional real vector space on which [|] is an inner
product of type ( + .+). Fix R > 0 and select a point a in one component
H of the hyperboloid
cfw_z V : [z |z ] = R2 .
Let B = BR (a ) be the ball

Dierential Geometry 2
Homework 01
1. Let V be a nite-dimensional real vector space on which [|] is an inner
product of type ( + .+). Fix R > 0 and select a point a in one component
H of the hyperboloid
cfw_z V : [z |z ] = R2 .
Let B = BR (a ) be the ball

Dierential Geometry 2
Homework 02
1. Let (M, g ) be a Riemannian manifold in which is a smooth curve. Show
that the operator of covariant dierentiation along
D : Vec Vec
may be dened by the requirement that if Vec and 1 (M ) then
(t) [(D )t ] =
d
[ (t)

Dierential Geometry 2
Homework 02
1. Let (M, g ) be a Riemannian manifold in which is a smooth curve. Show
that the operator of covariant dierentiation along
D : Vec Vec
may be dened by the requirement that if Vec and 1 (M ) then
(t) [(D )t ] =
d
[ (t)

Dierential Geometry 2
Homework 03
1. Let S V be the unit sphere in a Euclidean space. Recall that if is
V
S
a smooth curve in S then D = (D )T . Use this fact to determine the
(unit-speed) geodesics in S explicitly. Do the same for geodesics in hyperbolic

Dierential Geometry 1
Homework 02
1. Let M be a smooth manifold. By a smooth curve through p in M we mean
a smooth map : I M from some open interval (0 ) I R such that
(0) = p.
(a) Show that a tangent vector Tp M is dened by the rule
f C (M ) (f ) = (f )