MA 755 Fall 05. Notes #1. I. Kogan.
Manifolds and tangent bundles. Vector fields and flows.
1
Differential manifolds, smooth maps, submanifolds
Definition 1 An n-dimensional C k -differentiable manifold is a second countable, Hausdorff topological space together with a differentiable structure, given by a countable collection of open sets
Uα ⊂ M (where α ∈ A ⊂ N) and homeomorphisms ψα : Uα → Vα , where Vα is an open connected
subset of Rn , with the following conditions are satisfied:
• ∪α Uα = M ;
• For every Uα and Uβ such that Uα ∩ Uβ 6= ∅, let Wα = ψα (Uα ∩ Uβ ) ⊂ Vα and let Wβ =
ψβ (Uα ∩ Uβ ) ⊂ Vβ . Then the bijection ψβ ◦ ψα−1 : Wα → Wβ and its inverse are C k differentiable.
Open set Uα are called (coordinate) charts, maps ψα are called coordinate maps. Their collection
{(Uα , ψα )} is called atlas on M , or the differentiable structure on M , maps ψβ ◦ ψα−1 : Wα → Wβ are
called transition maps. From now on we will assume, unless it is explicitly stated otherwise, that
k = ∞ in the above definition, that is, the transition maps, are smooth. Such manifolds are called
smooth manifolds. If the transition maps are analytic, then the manifold is called analytic.
Problem 1 Cover a circle S 1 of radius one centered at the origin in R2 with two coordinate charts:
U1 = S 1 − {(0, 1)} and U2 = S 1 − {(0, −1)}. Write down explicitly (in coordinates) projections π1
from U1 to the line y = −1 and π2 from U2 to the line y = 1. Describe the sets W1 = π1 (U1 ∩ U2 )
and W2 = π1 (U1 ∩ U2 ) and compute the transition maps π2 π1−1 : W1 → W2 explicitly. Explain to
yourself why π1 and π2 are coordinate maps.
Problem 2 Provide similar covering for the the sphere S 2 ⊂ R3 with two coordinate charts, find
coordinate maps that correspond to projections from the north and south polls. Find transition
maps. Can you generalize this construction to an n-dimensional sphere S n ⊂ Rn+1 ?
As always in math, once a category of objects is defined (think of a linear space, a topological space,
a group, for example), we need to define what classes of objects inside this category are considered
to be the same (think of terms: isomorphic, homeomorphic etc) and what class of maps between the
objects it makes sense to consider (think of linear maps, continuous maps, group homomorphisms
etc).
Definition 2 Two smooth n-dimensional manifolds M, {(Uα , ψα )} and N, {(Wβ , ξβ )} are diffeomorphic if there is a bijection f : M → N such that ∀a ∈ M and ∀Uα 3 a and ∀Wβ 3 f (a), the
bijection ξβ ◦ f ◦ ψα−1 and its inverse are smooth maps.
Problem 3 Illustrate what is going on by a picture
1
Two diffeomorphic manifolds considered to be the same in the manifold theory. Assume that
M = N as sets and f is the identity map in the above definition. Does it mean that M, {(Uα , ψα )}
and M, {(Wβ , ξβ )} are diffeomorphic? In other words, is it true that the same topological space
can have essentially unique differential structure? The answer is no. More details are on p. 39
of Spivak, Differential Geometry v1. In particular it is proved that there is unique differential
structure on Rn for n 6= 4, but for n = 4 it is unsettled. There is unique differential structure
on S n (sphere of dimension n) for n ≤ 6, but there are 28 different diff. structures for n = 7 and
more than 16 million for n = 31. If M, {(Uα , ψα )} and M, {(Wβ , ξβ )} are diffeomorphic, then the
corresponding atlases are called compatible. The union of all charts of all atlases compatible with
M, {(Uα , ψα )} is called the universal atlas of M, {(Uα , ψα )}. Note that the universal atlas may
contain uncountable number of charts.
Definition 3 A map f from a manifold M, {(Uα , ψα )} to a manifold N, {(Wβ , ξβ )}, where dim M =
m and dim N = n, is smooth if ∀a ∈ M and ∀Uα 3 a and ∀Wβ 3 f (a) the map ξβ ◦f ◦ψα−1 : Rm → Rn
is smooth.
Problem 4 Explain why, for a given a, checking smoothness using a single chart on M containing
a and a single chart on N containing containing f (a), instead of all such charts is sufficient.
Definition 4 An immersed submanifold of M̃ is a manifold M together with an injective smooth
map ι: M → M̃ .
Definition 5 An submanifold ι: M → M̃ is embedded, or regular,if for every point p ∈ ι(M ) there
exists an open nighborhood of Ũ ∈ M̃ such that ι−1 (Ũ ∩ ι(M )) is a connected open subset of M .
Theorem 6 Assume that a manifold M is a subset of M̃ and the inclusion map ι is an embedding.
Then near every a ∈ M there exists chart Ũ ⊂ M̃ , with coordinate functions x1 , . . . , xm such that
Ũ ∩ M is given by {x|xn+1 = 0, . . . , xm = 0}.
Such coordinate system is called slice coordinates. See more on p.15 of the text-book.
Problem 5 A two dimensional sphere S 2 = {(x, y, z)|x2 + y 2 + z 2 = 1} is embedded in in R3 . Find
slice coordinates on the upper subset {(x, y, z)|z > 0 ∈ R3 }.
Example 7 Figure eight, defined by a map
φ(t) = (sin(2 arctan t), 2 sin(4 arctan t)),
is not an embedded but an immersed submanifold of R2 . The same image is obtained by
φ̃(t) = (− sin(2 arctan t), 2 sin(4 arctan t))
However, the map φφ̃−1 : R → R is not smooth.
2
2
Tangent spaces on a manifold.
Definition 8 A function f : M → R is smooth if ∀a ∈ M and for all charts (Uα , ψα ) such that
Uα 3 a the map f ◦ ψα−1 : Rn → R is smooth.
Definition 9 A smooth curve is a smooth mapping γ: R → M . More precisely if Uα is any coordinate chart which contains a point γ(t) for some t ∈ R then the map ψα ◦ γ: R → Rn is smooth.
Problem 6 Consider a sphere S 2 = {(x, y, z)|x2 + y 2 + z 2 = 1} ∈ R3 . Let γ(t) = (x(t), y(t), z(t))
be a curve on S 2 . What equation does the tangent to the curve γ(t) = (x0 (t), y 0 (t), z 0 (t)) satisfy?
S 2 is a two dimensional manifold, so it is more appropriate to use two coordinate functions. Use
a coordinate chart U1 which excludes north pole, find explicit projection π1 : U1 → R2 and compute
˙
equations of the curve γ̃ = π1 ◦ γ. Write explicitly the tangent vector γ̃.
Definition 10 Two curves γ1 : R → M and γ2 : R → M , such that γ1 (0) = γ2 (0) = a are said to
have the same tangent at a (or in other words to have the same first order contact at a) if for any
Uα 3 a curves φα ◦ γ1 : R → Rn and φα ◦ γ2 : R → Rn have the same tangent at the point φα (a).
Problem 7 In the context of Problem 6 show that two curves on the sphere S 2 ⊂ R3 have the
same tangent at a point a, according to Definition 10 if and only if they have the same tangent in
the usual sense as curves in R3 .
Problem 8 Note that a curve is defined to be a map R → M , and not just as the image of this map.
This exercise underscores that the image of a smooth curve can have cusps and self intersection,
that two curves with the same image have different tangents at the same point. In other words
parametrization matters!
1. Sketch the curves γ(t) = (x(t), y(t)) = (t2 , t3 ), and β(t) = (x(t), y(t)) = (t2 − 1, t3 − t) on R2 .
Are these curves smooth in the sense of definition 9? Does their images on R2 look smooth?
Compute vectors γ̇(0) and β̇(0). To which point of R2 are these tangent vector attached?
2. Sketch the curves β1 (t) = (x(t), y(t)) = (4t2 − 1, 8t3 − 2t) = ((2t)2 − 1, (2t)3 − (2t)) on R2 ,
compare it with the curve defined by β. Note that β and β1 are different curves according to
definition 9. Compute β̇1 (0). To which point of R2 is this tangent vector attached? Compare
it with β̇(0).
3. Sketch the curves β2 (t) = (x(t), y(t)) = (t − 1)2 − 1, (t − 1)3 −
(t
−
1)
= (t2 − 2t, (t − 1)(t2 −
2t)) and β3 (t) = (x(t), y(t)) = (t + 1)2 − 1, (t + 1)3 − (t + 1) on R2 , compare them with the
curve defined by β. Compute β̇2 (0) and β̇3 (0). To which point of R2 are these tangent vectors
attached?
Definition 11 The tangent space at a point a ∈ M (denoted T M |a ) is a set of equivalence classes
of curves, γ: R → M such that γ(0) = a. Two curves are in the same class if they have the same
tangent at a.
3
Problem 9 Define addition and scalar multiplication on the elements of T M |a . Show that T M |a
is isomorphic to Rn as a linear space.
T M |a can be imagined as a linear space Rn of vectors attached to the point a. Note that one can
not add vectors attached to two different points. A smooth map F between two manifolds induces
smooth maps between tangent spaces, called it differential of F .
3
Directional derivatives and derivations
For every smooth curve γ: R → M and a smooth function f : M → R the function f ◦ γ: R → R is
smooth.
Definition 12 The derivative of a function f in the direction of vector γ̇ ∈ T M |a at a point
a = γ(0) is defined by
d
γ̇f (a) = f (γ(t)).
dt t=0
It is not difficult to check that γ̇ satisfies the following two important properties, ∀f, g ∈ C ∞ (M )
smooth functions on M and c ∈ R:
γ̇(f + cg)(a) = γ̇f (a) + cγ̇g(a)
γ̇(f g)(a) = (γ̇f (a)) g(a) + f (a)γ̇g(a)
linarity,
(1)
Leibniz (product) rule.
(2)
An assignment of a tangent vector at every point a ∈ M , that varies smoothly from point to point
is called a a vector field on M (see Definition 16). A map C ∞ (M ) → C ∞ (M ), that satisfies the
above two properties is called a smooth derivation. There is a one-to-one correspondence between
vector fields on a manifold and derivations explored in MA 555 notes 2.
Remark 1 We often omit coordinate maps and identify a point a in M with its coordinates:
p = (x1 , . . . , xm ). The corresponding basis for T M |a consists of vectors E1 , . . . , Em , such that
Ei (xj ) = δij . It is natural to denote this basis as ∂x∂ 1 , . . . ∂x∂m (or simply ∂1 , . . . , ∂m ). A tangent
vectors X ∈ T M |a is written as X = ξ 1 ∂x∂ 1 + . . . + ξ n ∂x∂n . One need to be careful, however, with
what happens under the change of coordinates!!! Similarly we can identify a function f : M → R
∂f
n ∂f
with its coordinate presentation f (x1 , . . . , xm ). Then Xf (a) = ξ 1 ∂x
1 (a) + . . . ξ ∂xn (a)
Problem 10 a) Let γ(t) = (sin t cos t, cos2 t, sin t) be a curve on the unit sphere S 2 ⊂ R3 . Let
a = (0, 1, 0) ∈ S 2 and f : S 2 → R defined by f (x, y, z) = x + y + z. Compute γ̇f (a).
b) Use the map π1 obtained in Exersice 6 to find a curve γ̃(t) = π1 ◦ γ: R → R2 and the function
f˜ = f ◦ π1−1 : R2 → R. Compute b = π1 (a) ∈ R2 and γ̃˙ f˜(b). Compare you answer with the answer
in part a). Make a picture that relates parts a) and b) of the problem.
4
The differential of a map. Immersions and submersions.
Definition 13 Let F : M → N be a smooth map between manifolds. The differential of F at a is
a map F∗ |a : T M |a → T M |F (a) , defined as follows:
4
Assume that γ̇ ∈ T M |a is defined by a smooth curve γ: R → M , such that γ(0) = a. Then γ̃ = F ◦γ
˙ In local coordinates F∗ |a
is a smooth curve on N , such that γ̃(0) = F (a). We define F∗ |a (γ̇) = γ̃.
is given by the Jacobian of F at a.
Vector F∗ |a (γ̇) is called the pushforward of γ̇ under F . Assume dim M = m and dim N = n then
• If rank F∗ |a = m at every point then F is called local immersion. If F is also one-to-one it
is called immersion
• If rank F∗ |a = n at every point then F is called submersion
• If rank F∗ |a = n = m at every point then F is called local diffeomorphism. If F is also
one-to-one, then it is a diffeomorphism.
5
Tangent bundle
Definition 14 A tangent bundle T M of a manifold M is the union of tangent spaces at all point
of M , that is T M = ∪a∈M T M |a = {a, γ̇(a)}
Theorem 15 If M is smooth m-dimensional manifold, then T M is a smooth 2m-dimensional
manifold. Moreover T M has a differential structure, such that the surjection π: T M → M , given
by π(a, γ̇(a)) = a, is smooth.
proof: Main idea: one can show that if {Uα , φα } is an atlas on M then {∪a∈Uα T M |a , (φα , φα ∗ )}
is an atlas on T M . Definition 16 A smooth vector field on M is a smooth map X: M → T M such that π ◦ X = id.
Example 17 Let x1 , . . . , xn be the standard Cartesian coordinate functions on Rn . (Note that that
xi : Rn → R i = 1..n aresmooth
functions.) Given
a smooth curve γ(t) = (x1 (t), . . . , xn (t)) one
1
dxn obtains a vector γ̇(0) = dx
dt t=0 , . . . , dt t=0 attached to a point a = γ(0). The components of
this vector are real numbers. Thus if a ∈ Rn then the tangent space T Rn |a ∼
= Rn is a set of vectors
n
n
n
2n
∼
∼
attached to the point a. The tangent bundle T R = R × R = R
Problem 11 Find an atlas on the tangent bundle on T S 1 , where S 1 is a unit circle.
A tangent bundle is an example of a vector bundle. See p16 of the text-book for the precise
definition. A vector field is a section of this bundle. Both T Rn ∼
= Rn × Rn and T S 1 ∼
= S 1 × R are
2
2
2
trivial. T S is an example of non-trivial bundle (it is not diffeomorphic to S × R ).
6
Tangent vectors under a change of coordinates
Let (U1 , φ1 ) and (U2 , φ2 ) be two overlapping charts on an n-dimensional manifold M . Then φ2 ◦φ−1
1
is an invertible smooth map from an open subset V ⊂ Rn to an open subset W ⊂ Rn . Let x1 , . . . , xn
5
be coordinate functions on the first copy of Rn and y 1 , . . . , y n be coordinate functions on the first
copy of Rn . Then φ2 ◦ φ−1
1 defines an invertible differentiable map φ: V → W :
y1
=
y 1 (x1 , . . . , xn )
...
y
n
=
(3)
n
1
n
y (x , . . . , x )
Let γ(t) be curve on M , such that γ(0) = a ∈ U1 ∩ U2 . Let φ1 (a) = x and φ2 (a) = y. Then the
curves x(t) = φ1 ◦ γ(t) and y(t) = φ2 ◦ γ(t) define the coordinates of the tangent vector γ̇. Indeed
1
T
dxn it has coordinates ẋ = dx
,
.
.
.
,
(we arrange the coordinates in a column vector) in
dt t=0
dt t=0
T
1
dy n the first coordinate system, and coordinates ẏ = dy
,
.
.
.
,
in the second.
dt
dt t=0
t=0
Theorem 18
ẏ = Jφ (x)ẋ.
(4)
proof: We have y(t) = y 1 (x1 (t), . . . , xn (t)), . . . , y n (x1 (t), . . . , xn (t)) . Hence
!
n
n
X
X
∂y 1
dxi ∂y n
dxi ẏ(t)|t=0 =
(x)
,...,
(x)
∂xi
dt t=0
∂xi
dt t=0
i=1
i=1
Corollary 19



7
∂
∂y 1

..
.
∂
∂y n


−T 
 (Jφ ) 
∂
∂x1
..
.
∂
∂xn


,
(5)
Integral curves and flows of vector fields
Definition 20 The integral curve of the vector field V through a point a ∈ Rn on Rn is a curve
γ: R → Rn such that γ(0) = a and
dγ
= V (γ(t)).
(6)
dt
Let x1 , . . . xn be coordinate functions on Rn , then V = ξ 1 ∂x∂ 1 + . . . + ξ n ∂x∂n , where ξ i ,
smooth functions on Rn , and (6) becomes a system of the 1st order ODE’s
dx1
dt
dxn
dt
i = 1..n are
= ξ 1 (x)
(7)
..
.
(8)
= ξ n (x).
(9)
6
A classical theorem assures that there exists a unique smooth solution of this system with any
initial condition xi (0) = ai ∈ R, i = 1..n. Equivalently there exists a unique integral curve of
the vector field V through every point. Due to the uniqueness, the integral curves do not intersect.
Thus Rn is a disjoint union of integral curves of a vector field V . We say that integral curves form
a foliation of Rn . Note that if V (a) is the zero vector, then the integral curve through a is a point:
γ(t) = a, ∀t ∈ R.
Problem 12 Find the integral curve of the vector field V =
condition a) (x0 , y0 ); b) (0, 1).
∂
∂x
∂
+ x ∂y
on R2 with the initial
Definition 21 A smooth map Φ: R × Rn → Rn is called a flow of a vector field V if
•
dΦ
dt (t, x)
= V (Φ(t, x)).
• Φ(0, x) = x.
The existence of such map follows from the existence of the integral curve through each point.
Indeed, for every fixed point x0 ∈ Rn the map Φ(t, x0 ): R → Rn is a smooth integral curve of V
through x0 .
By fixing arbitrary value t0 ∈ R we obtain the smooth map Φt0 (x) = Φ(t0 , x) : Rn → Rn .
Theorem 22 The set of maps Φt , t ∈ R have the following properties:
• Φ0 = id (that is, Φ0 is the identity map: Φ0 (x) = x for all x ∈ Rn ).
• Φt1 ◦ Φt2 = Φt1 +t2 .
• Φ−t = Φt −1 . In particular, this means that every map Φt has a smooth inverse and therefore
is a diffeomorphism.
Corollary 23 Let Φ(t, x) be the flow of a vector field V . The set of maps {Φt : Rn → Rn } is a
one-parameter group of diffeomorphism isomorphic to the abelian group R.
∂
∂
+ x ∂y
on R2 compute the flow Φ(t, x). Check that the
Problem 13 For vector field V = ∂x
properties of Theorem 22 are satisfied. Compute the inverse of the diffeomorphism Φ2 .
8
Lie series and Lie derivatives of a function.
Theorem 24 Let f : Rn → R be an analytic function and V be an analytic vector field on Rn ,
whose flow is Φ(t, x). Then for all x ∈ Rn we have the following power series expansion along the
integral curve γ(t) = Φ(t, x), called the Lie series of f .
∞
f (Φ(t, x)) = f (γ(t)) = f (x) + tV f (x) +
X tk
t2
V (V (f )) (x) + . . . =
V k (f )(x) := etV f (x). (10)
2
k!
k=0
7
By V k f (x) we mean that the derivation V is applied k times to the function f , and the resulting
function V k f is then evaluated at a point x. For a fixed point x, the result is a number. The
last equality is just a natural abbreviation, motivated by Taylor expansion for the usual exponent
function.
∞ k
X
t k
etV :=
V .
k!
k=0
If we also introduce notation
form:
etV x0
= Φ(t, x0 ), then the Lie series can be written in a compact
f (etV x) = etV f (x).
An outline of the proof of theorem 22. Start with the usual Taylor series:
2
k
∞ k 2
X
d
d
d
t
t
f (γ(t)) +. . . =
f (γ(t))
f (Φ(t, x)) = f (γ(t)) = f (γ(0))+ f (γ(t)) +
dt
dt
k! dt
t=0 2
t=0
k=0
d
d
From the definition of the flow it follows that dt
f (γ(t))t=0
= dt f (Φ(t, x)) t=0 = V f (x). Use this
k
d
f (γ(t))
= V k f (x)
as the basis case to prove by induction that dt
t=0
From (10) it follows that
f (etV x) − f (x)
= V f (x).
t→0
t
Similar limits along the integral curves of V can be computed not only for functions, but also for
vector fields, differential forms and other tensor fields. Such limits are called Lie derivatives. Thus
the Lie derivative of f along the flow of V is simply the directional derivative V f .
lim
∂
∂
Problem 14 a) Given a vector field V = ∂x
+ x ∂y
on R2 , find the Lie derivatives and Lie series
for f (x, y) = x and g(x, y) = y. Compare the results with what you found in Problem 12. Also find
2
Lie derivatives and Lie series for f (x, y) = x2 + y 2 and g(x, y) = y − x2 . Which one of these two
functions is constant along any integral curve? Why?
∂
∂
b) Given a vector field V = −y ∂x
+x ∂y
on R2 , find the Lie derivatives and Lie series for f (x, y) = x
and g(x, y) = y. Could you find/guess a function that is constant along any integral curve?
8
.
t=0