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Theodore Voronov.
§2
Differentiable Manifolds. Spring 2011
Constructions of manifolds
Last updated: April 14, 2011.
2.1
Product of manifolds.
Let X and Y are manifolds of dimensions n and m, respectively. Then the
Cartesian product X × Y has a natural structure of a manifold of dimension
n + m.
Construction: if ϕ1α : V1α → U1α are charts for X and ϕ2µ : V2µ → U2µ
are charts for Y (here V1α ⊂ Rn , V2µ ⊂ Rm ), then
ϕ1α × ϕ2µ : V1α × V2µ → U1α × U2µ ⊂ X × Y
are charts for X × Y . Here
ϕ1α × ϕ2µ : (x1 , . . . , xn , y 1 , . . . , y m ) 7→ (x, y) ∈ X × Y ,
x = ϕ1α (x1 , . . . , xn ), y = ϕ2µ (y 1 , . . . , y m ) .
If (V1α , ϕ1α ) and (V2µ , ϕ2µ ) are smooth atlases, then a direct check shows that
the atlas (V1α × V2µ , ϕ1α × ϕ2µ ) is also smooth. Changes of coordinates on
the product X × Y have the typical “diagonal” form:
0
0
xa = xa (x1 , . . . , xn ) ,
0
0
y i = y i (y 1 , . . . , y m ) .
Here a = 1, . . . , n and i = 1, . . . , m. In the first line we see transformations
of coordinates from X, in the second line, from Y .
In the same way one can consider products of three or more manifolds:
X1 × . . . × X k .
Example 2.1. The product S 1 × S 1 is called the torus or more precisely
2-torus, and is diffeomorphic with the “surface of a bagel” in R3 (check!).
(The surface of a bagel is the surface of revolution of a circle of radius r and
center (R + r, 0) in the xy-plane, where R > 0, r > 0, about the z-axis.)
Notation: T 2 .
Example 2.2. For any n, the n-fold product S 1 ×. . .×S 1 is called the n-torus.
Notation: T n . (Using induction, T n can be visualized as a ‘multidimensional
surface of revolution’ of T n−1 in Rn+1 in the same way as T 2 is obtained by
revolving T 1 = S 1 in R3 .)
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Theodore Voronov.
Differentiable Manifolds. Spring 2011
Example 2.3. For any manifold M , the product M ×R is called an (infinite)
cylinder over M .
Example 2.4. If we replace R by the open interval (0, 1), we obtain a finite
cylinder M × (0, 1). (Sketch a picture for M = S 1 .)
Remark 2.1. The product of a manifold with a closed segment [0, 1] is NOT
a manifold. (Explain, why.) Actually, it gives an example of a “manifold with
boundary”, a notion to be discussed later.
2.2
2.2.1
Submanifolds. Specifying manifolds by equations
Open submanifolds
Consider a manifold M = M n . Let A = (ϕα : Vα → Uα ) be an atlas for M .
Which subsets S ⊂ M are naturally manifolds?
Definition 2.1. A subset S ⊂ M is called an open submanifold if for all α,
the subsets ϕ−1
α (S ∩ Uα ) ⊂ Vα are open.
Any open submanifold S ⊂ M has a natural structure of a smooth manifold of the same dimension as M .
Indeed, consider Ūα := S ∩ Uα , V̄α := ϕ−1
α (S ∩ Uα ) and ϕ̄α := ϕα |V̄α . By
the definition, the maps ϕ̄α give bijections V̄α → Ūα , and, since V̄α ⊂ Vα are
n
open subsets
maps ϕ̄α are n-dimensional charts for the set
S of Vα ⊂ R , the S
S. Since Uα = M , we have Ūα = S, so Ā = (ϕ̄α : V̄α → Ūα ) is an atlas
for S. The changes of coordinates for the atlas Ā are simply the restrictions
to open subsets of the changes of coordinates for the atlas A. Therefore they
are smooth. So the set S considered together with the atlas Ā is a smooth
manifold of dimension n.
Example 2.5. Every open set U ⊂ Rn is an open submanifold. The manifold
structure is given by the restriction of the standard coordinates x1 , . . . , xn
onto U .
Example 2.6. Consider the subset of all invertible matrices GL(n) ⊂ Mat(n) .
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Since GL(n) = {A ∈ Mat(n) | det A 6= 0} is an open subset of Mat(n) ∼
= Rn ,
it is therefore an open submanifold and dim GL(n) = n2 .
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Theodore Voronov.
Differentiable Manifolds. Spring 2011
Remark 2.2. The manifold GL(n) is called the general linear group. It is a
group w.r.t. the matrix multiplication and all the group operations are given
by smooth maps. Groups that are also manifolds and satisfy this condition
are known as Lie groups. They make a very important class of manifolds.
Example 2.7. For an arbitrary function f ∈ C ∞ (M ), the subset
Mf = {x ∈ M | f (x) 6= 0}
is an open submanifold. Indeed, for each coordinate chart, the set ϕ−1
α (Mf ∩
Uα ) it open because it is the preimage of the open set R \ {0} under the
smooth (hence, continuous) function fα = f ◦ ϕα : Vα → R.
Later, when we introduce topology on manifolds, we shall see that open
submanifolds of M are, in particular, open sets in M .
2.2.2
Closed submanifolds
Consider now a different kind of submanifolds.
Definition 2.2. A subset S ⊂ M of a manifold M = M n is called a closed
submanifold of dimension k if in some atlas A on M , for all α,
k+1
ϕ−1
= . . . = xn = 0} ∩ Vα .
α (S ∩ Uα ) = {x
Denote Ūα := S ∩ Uα , V̄α := ϕ−1
α (S ∩ Uα ) and ϕ̄α := ϕα |V̄α . We obtain
an atlas for S. The changes of coordinates have the form:
xi(α) = ϕiαβ (x1(β) , . . . , xk(β) , 0 . . . , 0) ,
i = 1, . . . , k ,
where ϕiαβ for i = 1, . . . , n, are the changes of coordinates for the atlas A.
Therefore S is a smooth manifold of dimension k.
The number n − k is often referred to as the codimension of S k in M n .
Definition 2.2 may look too restrictive because of a very special appearance of a closed submanifold in coordinate charts. In fact, it is not. We shall
see that closed submanifolds can be equivalently described in a less restrictive
way. For this, we need to analyze systems of equations in Rn .
Consider a system of p equations for n variables where p 6 n :
 1 1
n

 f (x , . . . , x ) = 0
...
(1)

 f p (x1 , . . . , xn ) = 0
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Theodore Voronov.
Differentiable Manifolds. Spring 2011
where the LHS’s are smooth functions on Rn . Let S ⊂ Rn be the solution set
of (1), i.e., the set of all points of Rn satisfying equations (1). We say that
equations (1) are independent if the rows of the matrix of partial derivatives
 ∂f 1 ∂f 1

∂f 1
.
.
.
1
2
n
∂x
∂x
∂x
∂f
,
= . . .
(2)
∂x
∂f p
∂f p
∂f p
. . . ∂xn
∂x1
∂x2
which is a function of a point in Rn , are linearly independent atall points
∂f i
x ∈ Rn belonging to the subset S ⊂ Rn . That means that rank ∂x
=p
j
at all points x ∈ S. If p = 1, i.e., for a single equation
f (x1 , . . . , xn ) = 0 ,
(3)
we use ‘non-degenerate’ instead of ‘independent’, so we say that equation (3)
is non-degenerate if df (x) 6= 0 for all points of the solution set.
Recall that the rank of a matrix is the dimension of the vector space
spanned by its rows — the row space — or the dimension of the vector space
spanned by its columns —the column space. These two numbers coincide
and also coincide with the maximal size of an invertible square submatrix,
i.e., the maximal size of a non-zero minor.
We may consider equations such as (1) or (3) defined on open domains
of Rn and apply the same notions of independence or non-degeneracy.
Lemma 2.1. Consider an open domain V ⊂ Rn . Let Γ ⊂ V be specified by
independent equations f i (x) = 0, where i = 1, . . . , p. Let k = n − p. Then for
every point x0 ∈ Γ there is an open set W ⊂ V , x0 ∈ W , and new curvilinear
coordinates y 1 , . . . , y n on W such that Γ ∩ W is specified by the equations
y k+1 = . . . = y n = 0.
= p means that at each x0 ∈ Γ there is an
Proof. The condition rank ∂f
∂x
invertible p × p submatrix of the rectangular matrix ∂f
. Fix some x0 ∈ Γ.
∂x
Without loss of generality, we may assume that it is the submatrix
i
∂f
∂xj i,j=k+1,...,n
consisting of the last n − k = p columns that is invertible on some open
neighborhood W 0 of x0 . (Otherwise renumber the coordinates.) By the
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Theodore Voronov.
Differentiable Manifolds. Spring 2011
implicit function theorem, we can resolve the equations w.r.t. xi , i > k + 1,
on a smaller neighborhood W ⊂ W 0 of x0 . So there are smooth functions
g i (x1 , . . . , xk ) on W such that the equations f i (x) = 0 are equivalent to
xi = g i (x1 , . . . , xk ) for i > k. Define new coordinates on W :
(
xi
if i = 1 . . . , k ,
yi =
i
i 1
k
x − g (x , . . . , x )
if i = k + 1 . . . , n .
This an invertible change of coordinates, and the subset Γ ∩ W is specified
by y j = 0 for all j > k + 1 as desired.
Corollary 2.1. A system of p independent equations on Rn specifies a closed
submanifold Γk ⊂ Rn of dimension k = n − p.
Proof. Use the notation of the lemma. We can consider a smooth atlas on Rn
constructed as follows. As coordinate domains Uα we take all neighborhoods
W for all points x0 ∈ Γ, with the corresponding coordinate systems (y i ),
as constructed in the lemma, and, in addition, the open set Rn \ Γ with
the standard coordinates (xi ). It is clear that Definition 2.2 is satisfied for
Γ ⊂ Rn if we use this atlas.
Corollary 2.2. For an arbitrary manifold M n , a subset S ⊂ M n is closed kdimensional submanifold if and only if for some atlas A = (ϕα : Vα → Uα ),
n
the subsets ϕ−1
α (S ∩ Uα ) ⊂ Vα ⊂ R (for all α) can be specified by n − k
independent equations.
Proof. Consider a subset S ⊂ M . If it satisfies Definition 2.2, it is obviously specified by independent equations in each chart, because the equations
xk+1 = 0 , . . . , xn = 0 are independent. Conversely, suppose S, for some atlas
A, is specified by n−k independent equations in each coordinate chart. Then
we can apply Lemma 2.1 to each such chart and introduce smaller open subsets of Rn and the corresponding smaller subsets of M making a new smooth
atlas of M so that for the new atlas, the appearance of S in each coordinate
chart will be as required by Definition 2.2.
(In the proof of Corollary 2.2 we have implicitly used the following argument: if the intersection of a subset S ⊂ M with each coordinate domain
U ⊂ M is a closed submanifold of U , then S is a closed submanifold of M .)
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Theodore Voronov.
Differentiable Manifolds. Spring 2011
Remark 2.3. Considerations in this subsection are modeled on a linear
example. Let p 6 n. Consider a homogeneous linear system
Ax = 0
(4)
where x ∈ Rn and A is an p × n matrix. Equations (4) are independent if
the p rows of the matrix A are linearly independent. As we know, in this
case the solution set for (4) is a vector space of dimension n − p. Recall that
if we solve this system by the Gauss elimination, we obtain n − p variables
as ‘free variables’ (so that they can be set to any values), and the remaining
p variables expressed as linear functions of the free variables, and this gives
the general solution.
2.2.3
Specifying manifolds by equations in RN
We can use the above theory as a source of examples of manifolds.
Example 2.8. Consider once again the sphere S 2 defined as the unit sphere
in R3 . It is specified by the equation
x2 + y 2 + z 2 = 1 ,
where we used traditional notation for coordinates on R3 . In other words,
we have a single equation
f (x, y, z) = 0
where f (x, y, z) = x2 + y 2 + z 2 − 1. The set of solutions S is our sphere S 2 .
The matrix of partial derivatives is the row matrix
∂f ∂f ∂f
,
,
= (2x, 2y, 2z) .
∂x ∂y ∂z
We see that this row vector never vanishes for (x, y, z) 6= 0, in particular, it
does not vanish for all (x, y, z) ∈ S. Therefore our equation is non-degenerate
and Corollary 2.1 applies. Once again we obtain a manifold structure of S 2 ,
which, as one can easily see, is equivalent to that previously defined with the
help of stereographic projections. The point in Corollary 2.1 is the possibility
to solve the system in question w.r.t. some of the variables so that the
remaining variables are ‘free’ and can be taken as coordinates, thus giving
a manifold structure, which is deduced from the implicit function theorem.
For the equation
of the sphere we, of course, can do that directly, writing,
p
2
e.g., z = 1 − x − y 2 in a neighborhood of some point.
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Theodore Voronov.
Differentiable Manifolds. Spring 2011
In practical situations it often happens that interesting objects are defined
by equations that are not independent, and constructing an equivalent system
of independent equations may be awkward, if at all possible. (It is always
possible locally, of course, in a neighborhood of each point, but this is of no
practical help.) The following generalization of the previous is helpful.
Consider once again a system of p simultaneous equations on Rn :
 1 1
n

 f (x , . . . , x ) = 0
...
(5)

 f p (x1 , . . . , xn ) = 0
where, unlike (1), we do not assume that p 6 n. By definition, the rank of
this system is the rank r = rank ∂f
of the matrix of partial derivatives. It is
∂x
a function of a point x. We shall consider it at the points of the solution set
S specified by (5). The rank obviously cannot exceed n (and p). Suppose
rmax is the maximal value of the rank, which is attained at some point x0 .
Then the rank equals rmax on a whole open neighborhood of x0 . (Indeed,
the rank of a matrix is the maximal size of the non-zero minors. Therefore
a small variation of the matrix can only increase the rank, so if it is already
maximal, it will remain constant.) Hence the rank of a system of equations
is a constant and equals to rmax on a dense open subset of S ⊂ Rn . It makes
sense to concentrate on the case when it is simply constant.
Definition 2.3. We say that a system of equations (5) has constant rank if
at all points of the solution set S specified by (5), the rank of the matrix of
partial derivatives (2) is a constant number r (the same for all points x ∈ S).
A particular case is of course the case of independent equations treated
in Lemma 2.1 and Corollary 2.1, where r = p.
Theorem 2.1 (Manifolds specified by equations of constant rank). The solution set S of a system of constant rank on Rn is a closed submanifold of
Rn of dimension k = n − r where r is the rank of the system.
Proof. Omitted. It is similar to that of Lemma 2.1, but more technically
complicated.
How do we practically know that the rank of a given system is constant?
We should look at the vector space of solutions of the following auxiliary
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Differentiable Manifolds. Spring 2011
Theodore Voronov.
linear system:
X ∂f µ
i
∂xi
(x)ẋi = 0 .
(6)
Here x is a given point in the set of solutions. The unknowns are the variables
that we have denoted by ẋi . The matrix of the system is exactly the matrix
of partial derivatives for which we want to know whether its rank is constant
(independent of x) or not. From linear algebra we know that the dimension
of the space of solutions of (6) is the number of unknowns (which is n) minus
the rank of the matrix (which is r). Therefore the rank r of the matrix
∂f µ
(x) is constant for x ∈ S if and only if the dimension of the space of
∂xi
solutions of the auxiliary system (6) does not depend on x ∈ S.
Notice that the linear system (6) can be obtained by the formal differentiation of the system (5) w.r.t. a parameter t (“time”) so that the ‘dot’
stands for the “time derivative”. Later we shall see the geometrical meaning
of this.
Example 2.9. Recall that a matrix A is called orthogonal is AAT = E,
where E is the identity matrix. The set of all real orthogonal n × n matrices
is denoted O(n). One can check that O(n) is a group, called the orthogonal
group. We claim that O(n) has a natural manifold structure. Let us apply
Theorem 2.1. Notice that a single matrix equation such as AAT = E is in
fact a system of n2 equations for the n2 matrix entries. Obviously it cannot
be independent, otherwise it would specify a single point (at best). To see
that this system has constant rank we consider A as a function of a parameter
t, A = A(t), and differentiate the equation w.r.t. t. We obtain the auxiliary
linear system ȦAT + AT Ȧ = 0 or (ȦAT )T = −ȦAT . (Here A ∈ O(n) is a
fixed point of the solution set and the unknown is Ȧ.) The space of solutions
of the auxiliary linear system (for all A ∈ O(n)) is therefore isomorphic
to the space of all skew-symmetric n × n matrices. Hence it has constant
dimension n(n − 1)/2 independent of A and the rank of the original system
is constant. The dimension of the manifold is n(n − 1)/2. (The dimension of
the space of all skew-symmetric n × n matrices can be counted as the number
of independent entries for such a matrix: 21 (n2 − n), where n is the number
of the diagonal entries, which all have to be zero.)
Example 2.10. A particularly simple special case is the group O(2) of the
orthogonal 2 × 2 matrices. The equation AAT = E for n = 2 can be solved
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Differentiable Manifolds. Spring 2011
Theodore Voronov.
explicitly (do this!). Any orthogonal 2 × 2 matrix is either
cos α − sin α
A=
sin α cos α
or
A=
− cos β sin β
sin β cos β
.
(Geometrically it is either the rotation through some angle α or a rotation
followed by the inversion in a coordinate axis.) As expected we obtain an
example of a one-dimensional manifold: 2(2 − 1)/2 = 1. Moreover, we see
that O(2) ∼
= S 1 ∪ S 1 (the disjoint union of two circles).
Slightly changing the notation, consider RN and closed submanifolds in
it. One can prove that every manifold1 M n of dimension n is diffeomorphic
to a closed submanifold of RN for a sufficiently large N n. This should
not be mistakenly understood as that every manifold can be specified by a
system of equations on RN given by globally defined functions. It is true
only for special manifolds such as the sphere S n or the Lie groups such as
O(n). Being a closed submanifold of RN only means that M n ⊂ RN can be
defined by a system of equations of rank N − n in a neighborhood of each
point x ∈ M n , i.e., only locally (and when we pass to a different point, the
system, generally, changes).
1
More precisely, certain general topological conditions should be satisfied by M n for
this to be true. These conditions are normally included in the definition of a manifold and
we discuss them in the next section.
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