Differential Manifolds
Chapter I: Sheaves and Differential Manifolds:
Definitions and Examples
3 Dierential Manifolds
After these preliminaries, we are now ready to dene the concept of dierential man-
ifolds. These objects provide the proper setting for developing dierential and integral
calculus. In Physics, these are called conguration spaces and may be thought of as the
set of all possible states of the system of which one wishes to study the dynamics.
3.1. Denition. A dierential manifold M (of dimension n) consists of
a) a topological space which is Hausdor and admits a countable base for open sets,
and
b) a sheaf A
M
= A of subalgebras of the sheaf of continuous functions on M.
These are required to satisfy the following local condition. For any x ∈ M , there is an
open neighbourhood U of x and a homeomorphism of U with an open set V in R
n
such
that the restriction of A to U is the inverse image of the sheaf of dierentiable functions
on V .
The homeomorphisms referred to above, dened in a neighbourhood of any point,
are called coordinate charts. This is because composing with this homeomorphism the
coordinate functions in R
n
, one obtains functions x
1
, . . . , x
n
. We think of M as a global
object on which action takes place, and it is locally described by using coordinates.
3.2. Examples.
1) Since the concept of dierential manifolds is based on the notion of dierentiability
in R
n
, it is clear that R
n
together with the sheaf of dierentiable functions is a
dierential manifold. Slightly more abstractly, any nite-dimensional vector space
over R is a dierential manifold.
2) If (M, A) is a dierential manifold, and U an open subset of M , then the subspace
U, together with the restriction A|U of A is a dierential manifold as well. This
will be referred to as an open submanifold of M .
3) Combining these two examples, we see that any open subspace of R
n
(or any
nite-dimensional vector space) is a dierential manifold. In particular, the space
GL(n, R) of invertible (n, n)-matrices, which is actually the open set in the vector
space of all (n, n)-matrices given by the nonvanishing of the determinant, is a man-
ifold of dimension n
2
. Incidentally, it is also a group under matrix multiplication
and is called the General Linear group.
4) Let f be a dierentiable function in R
n
. Consider the closed subspace (the zero
locus of f) of R
n
given by
Z
f
= {x ∈ R
n
: f(x) = 0}
20
Section 3: Differential Manifolds
Then Z
f
has a natural structure of dierential manifold, if at every x ∈ Z
f
, at least
one of the partial derivatives of f does not vanish. In fact, consider the association
to any open set U ⊂ Z
f
, of the set of functions on U , which can be extended to
a dierentiable function on a neighbourhood of U in R
n
. This gives a presheaf
on Z
f
. (Actually it will turn out that it is a sheaf, but we do not need it here.)
Let A
Z
f
be the associated sheaf. Then by our assumption, for any x ∈ Z
f
, there
exists a neighbourhood N of x in R
n
such that, one of the partial derivatives, say
∂f
∂x
n
, is nonzero. By the implicit function theorem, the projection to R
n−1
taking
(x
1
, . . . , x
n
) to (x
1
, . . . , x
n−1
) is a dierentiable isomorphism of N ∩ Z
f
with an
open set V in R
n−1
. (that is to say, a dierentiable bijective map from N onto
V whose inverse is also dierentiable). It is now clear that this isomorphism gives
the local requirement of the sheaf A
Z
f
. If we further specialise the function to be
f(x) =
∑
x
2
i
− 1, then Z
f
is the set of vectors of unit length in R
n
which we call
the unit sphere S
n−1
.
On the contrary, if f is taken to be the function xy dened in R
2
with coordinates
x, y, then clearly it does not satisfy the criterion given above. In fact, both the
partial derivatives,
∂f
∂x
and
∂f
∂y
vanish at (0, 0). So we cannot conclude that Z
f
is
a dierential manifold in this case. In fact, the topological space Z
f
cannot have
a structure of a dierential manifold, since it is easy to see that no neighbourhood
of 0 in Z
f
is homeomorphic to an open interval in R.
5) The above example can be generalised further, by taking, instead of one function,
nitely many functions. So, let f = (f
i
), 1 ≤ i ≤ r, be nitely many dierentiable
functions in R
n
. Then the closed subspace
Z
f
= {x ∈ R
n
: f
i
(x) = 0 for all i}
has a natural structure of a dierential manifold, if f satises the following con-
dition. For every x ∈ Z
f
, the rank of the (r, n) matrix
(
∂f
i
∂x
j
)
is r. One may think
of f as a function into R
r
.
We will now take for f the function on (n, n)-matrices with values also in R
n
2
given by A 7→ AA
′
−I
n
, where A
′
denotes the transpose of A. Then one can check
that the above criterion is fullled and hence the topological space {A ∈ M
n
(R) :
AA
′
= I
n
} is a dierential manifold. This is actually a subgroup of GL(n, R) as
well, called the orthogonal group and is usually denoted O(n, R). Similarly take
the map f : M (n, C) into itself given by A 7→ AA
′
−I
n
, and get the set f = 0 as a
subgroup of GL(n, C). This group is called the unitary group and denoted U (n).
(Are these manifolds connected?)
6) Consider the real projective space RP
n
dened to be the quotient of the unit
21
Chapter I: Sheaves and Differential Manifolds:
Definitions and Examples
sphere S
n
by the identication of antipodal points x and −x. We may dene a
sheaf on RP
n
by associating to any open set U ⊂ RP
n
, the algebra of dierentiable
functions on its inverse image in S
n
which are invariant under the antipodal map.
For any point a ∈ S
n
, the open neighbourhood of a consisting of those y ∈ S
n
whose distance from a is less than 1 is mapped homeomorphically on an open set
in RP
n
, and it is easy to see that this takes the sheaf of dierentiable functions
on S
n
isomorphically onto the sheaf dened above. Thus we see that RP
n
is a
dierential manifold in a natural way.
3.3. Exercises.
1) Show that the topological subspace of the space of all (n, n)-matrices, consisting
of those matrices whose determinants are 1, is a dierential manifold. This is also
a subgroup of the group GL(n, R) mentioned above and is denoted SL(n, R). It
is called the Special Linear group.
2) Is the same true of matrices with determinant 0?
3.4. Glueing up dierential manifolds. Often, a structure of a dierential mani-
fold is given on a Hausdor topological space M with a countable base for open sets, by
the following procedure. Suppose {U
i
} is an open covering, and that each U
i
is provided
with a subsheaf A
i
of the sheaf of continuous functions making (U
i
, A
i
) a dierential
manifold. If we can glue all these sheaves together to get a sheaf of algebras on M,
then it is clear that it would make M a dierential manifold. We have already seen 2.5
how we can glue them together. What we need is simply that A
i
|U
i
∩U
j
is the same as
A
j
|U
j
∩U
i
. In other words, the open submanifold U
i
∩U
j
of U
i
is the same as the open
submanifold U
i
∩ U
j
of U
j
.
In particular, if (V
i
, A
i
) are open submanifolds of R
n
, then the glueing data may also
be formulated as follows. The space M is covered by open sets U
i
. For each i, one is
given a homeomorphism c
i
of U
i
with the open subset V
i
of R
n
. If U
i
and U
j
intersect,
then U
i
∩ U
j
has as images in V
i
and V
j
, two open sets which we may call V
ij
and V
ji
.
Then c
j
◦ c
−1
i
gives a homeomorphism V
ij
→ V
ji
. One may use the homeomorphism
c
i
to transport the sheaf of dierentiable functions on V
i
to a sheaf of algebras on U
i
.
But in order to glue these together, we need to know that its restriction to U
i
∩ U
j
is
the same as the restriction of the transported sheaf on U
j
. This can be achieved if and
only if the above homeomorphism V
ij
→ V
ji
is dierentiable for every i, j. (Note that
the inverse is also dierentiable, by reversing the roles of i and j.) This is in fact the
traditional denition of a dierential manifold.
For example, consider the sphere in R
n
. The map
(
x
1
, . . . , x
n
)
7→
(
x
1
, . . . ,
ˆ
x
i
, . . . , x
n
)
22
Section 3: Differential Manifolds
is a homeomorphism of the open set consisting of points of the sphere for which x
i
> 0
onto the open unit ball in R
n−1
. We can therefore provide these open sets with the
dierential manifold structure of the unit ball. We could do the same with the open
sets of the sphere in which x
i
< 0. It is clear that all these open sets, as i varies, cover
S
n−1
. To glue these up, we have only to check that the map
(
y
1
, . . . , y
n−1
)
7→
(
y
1
, . . . ,
√
1 − |y|
2
, . . . , ˆy
j
, . . . , y
n−1
),
where the term
√
1 − |y|
2
occurs at the i-th place, is a dierentiable isomorphism of the
open set of the unit ball onto the open ball.
Another example is provided by the complex projective space. Notice rst that
the real projective space RP
n
may also be dened as the quotient of R
n+1
\ {0} by
the equivalence relation: (x
0
, x
1
, . . . , x
n
) ∼ (y
0
, y
1
, . . . , y
n
) if and only if there exists a
nonzero real number a such that y
i
= ax
i
, for all i. An analogous denition makes sense
over complex numbers as well. In other words, dene the complex projective space CP
n
to be the quotient of C
n+1
\ {0} by the equivalence relation:
(z
0
, . . . , z
n
) ∼ (z
′
0
, . . . , z
′
n
)
if and only if there exists a ∈ C
×
such that z
′
i
= az
i
for all i. Since for any such
z = (z
0
, . . . , z
n
) at least one z
i
is nonzero, CP
n
is covered by open sets which are
images of sets U
i
, i = 0, . . . , n under the natural map C
n+1
\ {0} → P
n
, where U
i
=
{z = (z
0
, . . . , z
n
) : z
i
6= 0}. It is clear that the subspace z
i
= 1 of U
i
is mapped
homeomorphically onto the image of U
i
in CP
n
. On the other hand, it is homeomorphic
to C
n
under the projection which omits the i-th coordinate. This can be used as above
to put a structure of a dierential manifold on it. To check the condition for glueing up,
we only have to check that the map
{(z
0
, . . . , z
n
), z
i
= 1 , z
j
6= 0 } 7→ {(z
0
, . . . , z
n
), z
i
6= 0 , z
j
= 1 }
23
Chapter I: Sheaves and Differential Manifolds:
Definitions and Examples
given by (z
0
, . . . , z
n
) 7→ (z
0
/z
j
, . . . , z
n
/z
j
) is a dierentiable isomorphism. This is of
course obvious.
A little more abstractly, we could have replaced C
n+1
by any complex vector space
V of dimension n + 1. Then one considers the open set V \ {0} and introduces the
equivalence relation: v ∼ v
′
if and only if there exists a ∈ C
×
such that av = v
′
. The
quotient is dened to be the projective space P (V ) associated to V . If f is any nonzero
linear form on V , the set {v ∈ V : f (v) = 1} is mapped homeomorphically onto an open
set X
f
of P (V ). What we did above amounts to using this homeomorphism to dene
a dierential structure on X
f
. We may glue all these structures together to obtain a
dierential structure on the whole of P (V ).
One may also think of the points of P (V ) as one-dimensional subspaces of V . Then
one may generalise this by considering all r-dimensional vector subspaces of V for any
xed r < dim(V ). The set thus formed is called the Grassmannian of r-dimensional
subspaces of V . We will indicate two ways in which one can provide this set with the
structure of a dierential manifold. Firstly, let W be any such subspace. Consider the
image of the one-dimensional space Λ
r
(W ) in Λ
r
(V ). Thus to every element of the
Grassmannian we have associated an element of the projective space P (Λ
r
(V )). Then
one checks the following assertion.
3.5. Lemma. If ω is any nonzero element of Λ
r
(V ), then the linear map V → Λ
r+1
(V )
given by v 7→ v ∧ω has kernel of dimension ≤ r. Moreover, the kernel is of dimension r
if and only if ω ∈ Λ
r
(W ) for some r-dimensional subspace W of V .
Proof. In fact, it is obvious that if ω belongs to Λ
r
(W ), then every element of W is in
the kernel of the above map. Let {e
i
}, 1 ≤ i ≤ n, be a basis of V such that the rst r of
these generate W . In other words, e
1
∧ ··· ∧ e
r
can be taken to be ω. Let v =
∑
a
i
e
i
;
then v ∧ ω = 0 if and only if a
i
= 0 for all i > r. Thus the kernel of the map v 7→ v ∧ ω
is precisely W .
Conversely, if the kernel contains an r-dimensional subspace W , again take a basis
like the one above. Writing out ω in terms of a basis as
∑
i
1
<···<i
r
a
i
1
,...,i
r
e
i
1
∧···∧e
i
r
,
we deduce that if e
j
∧ ω = 0, then a
i
1
,...,i
r
= 0 whenever j does not belong to the set
{i
1
, . . . , i
r
}. Since we have assumed that e
j
∧ ω = 0 for all j ≤ r, we see that the only
nonzero coecient in the expression for ω is a
1,...,r
, that is to say, ω ∈ Λ
r
(W ). This also
shows in particular that the dimension of the kernel is ≤ r.
The above lemma asserts in fact that the Grassmannian is imbedded in P (Λ
r
V ) as a
closed subset. Using this description, one can check that the Grassmannian is actually
a closed submanifold. This imbedding is called the Plücker imbedding.
Another way of introducing the structure of dierential manifold on the Grassman-
nian is the following. In order to introduce a coordinate system in a neighbourhood of
any r-dimensional subspace U
0
of V , we proceed as follows. Fix an (n −r)-dimensional
24
Section 3: Differential Manifolds
subspace W of V supplementary to U
0
. Consider the set of all r-dimensional subspaces
U of V which are supplementary to W . This set evidently contains U
0
. The projection
of V onto W corresponding to the direct sum decomposition V = U ⊕ W restricted to
U
0
gives a linear map A
U
of U
0
into W . When U = U
0
this map is the zero map. In
general the linear map determines U as the image of η − A
U
, where η is the inclusion
of U
0
in V . This sets up a bijection between Hom(U
0
, W ) and the above set, thereby
coordinatising the Grassmannian in a neighbourhood of U
0
.
3.6. Denition. If (M, A) is a dierential manifold, then sections of A over an open
set U of M are called dierentiable functions on U.
Since A is a sheaf, the notion of a dierentiable function on M is of a local nature,
so that the following properties are obvious.
i) If f is a nowhere vanishing dierentiable function, then so is 1/f.
ii) If (U
i
) is a locally nite covering and (f
i
) is a family of dierentiable functions
with support in U
i
, then
∑
f
i
is also a dierentiable function.
iii) If φ is a dierentiable function on R
m
and f
1
, . . . , f
m
are dierentiable functions
on M, then φ(f
1
, . . . , f
m
) is also a dierentiable function.
3.7. Denition. A continuous map f of a dierential manifold M into another
dierential manifold N is said to be dierentiable if for any x ∈ M , and for every
dierentiable function φ in a neighbourhood U of f(x) in N , the composite φ ◦ f is a
dierentiable function on f
−1
(U).
From the denition of a dierentiable map it follows that there is a homomorphism
of the sheaf A
N
into f
∗
(A
M
) and therefore also a homomorphism of A
M
into f
−1
(A
N
).
This is called the structure homomorphism associated to f.
3.8. Remark. It is clear from the denition above that if M, N, P are dierential
manifolds, and f : M → N , g : N → P are dierentiable maps, then the composite
g◦f : M → P is also dierentiable. However, if f : M → N is a dierentiable map which
is bijective, then one cannot conclude that the inverse map is also dierentiable. The
hackneyed counterexample is the function x 7→ x
3
of R into R, which is dierentiable
and bijective, but whose inverse x 7→ x
1
3
is not dierentiable at 0.
3.9. Denition. A dierentiable map f : M → N of dierential manifolds is said to
be a dieomorphism if there is a dierentiable inverse.
A fact, basic to the study of dierentiable functions, is that there are lots of them.
We will now formulate this precisely.
Dierentiable partition of unity.
3.10. Proposition. The following are equivalent.
25
Chapter I: Sheaves and Differential Manifolds:
Definitions and Examples
i) Given any locally nite open covering (U
i
)
i∈I
of M, there exist dierentiable func-
tions φ
i
on M with values in the closed interval [0, 1] such that the support of φ
i
is contained in U
i
and
∑
φ
i
= 1 .
ii) Given open sets U, V with V ⊂ U , there exists a positive dierentiable function
whose support is contained in U and which does not vanish anywhere in V .
Proof. i) implies ii). Consider the open covering (M \ V , U ). By i), there exist [0, 1]-
valued functions φ, ψ with supports respectively in M \ V and U such that φ + ψ = 1
everywhere. Then ψ satises the requirement in ii).
ii) implies i). Let (V
i
) be a shrinking of (U
i
), i.e. an open covering with V
i
⊂ U
i
for
all i. By Assumption ii), we see that there exist positive dierentiable functions φ
i
such
that suppφ
i
⊂ U
i
and φ
i
are nonzero everywhere on V
i
. Now since the covering is locally
nite, the sum φ =
∑
φ
i
makes sense, is dierentiable and is nonzero everywhere. The
family of functions {ψ
i
= φ
i
/φ}, satises i).
Of course the point of the above proposition is that the equivalent properties stated
there are actually true. We will now prove this fact. Firstly, in order to prove ii),
it is enough to do it locally in the following sense. For every m ∈ M there exists a
neighbourhood N
x
such that there is a function as in ii) with V replaced by V ∩ N
x
and U by U ∩ N
x
. Then one replaces the sets N
x
by a locally nite renement of
notes that the sum of the corresponding functions fulls the requirement. Taking N
x
to be a coordinate neighbourhood of x, we therefore reduce the problem to proving the
following.
3.11. Proposition. Let S
1
, S
2
be concentric spheres in R
n
centered at 0, with S
1
⊂
S
2
. Then there exists a dierentiable function which is nonzero everywhere inside S
1
and has support contained in the interior of S
2
.
Proof. Clearly it is enough to construct a dierentiable function on R
n
which is nonzero
everywhere inside the unit ball and zero in the complement. The function x 7→ exp
(
1
∑
x
2
i
−1
)
for all x inside the unit ball and 0 in the complement, is such a function.
3.12. Denition. Let (U
i
) be a locally nite open covering of a dierential manifold
M. A family (φ
i
) of dierentiable functions on M with values in [0, 1] is said to be a
partition of unity with respect to the covering (U
i
), if the support of φ
i
is contained in
U
i
for every i and
∑
φ
i
= 1 .
We will now derive a simple consequence of the existence of a partition of unity.
3.13. Proposition. Any section of A over a closed set can be extended to a dieren-
tiable function on M. In other words, given a dierentiable function φ in a neighbour-
hood of a closed set K, there exists a dierentiable function ˜φ on M which coincides
with φ in a neighbourhood of K.
26
Section 3: Differential Manifolds
Proof. In fact, if K ⊂ U , and f is a dierentiable function on U , consider the partition
of unity with respect to the covering U, M \ V , where V is a neighbourhood of K
with V ⊂ U . Thus there is a dierentiable function φ on M which is 1 on V and
with support in U. The function fφ on U has support contained in the support of φ.
Hence the function fφ on U and the constant function 0 on M \ (supp f) coincide on
the intersection, thereby giving rise to a dierentiable function on the whole of M as
required.
3.14. Proposition. Let M, N be dierential manifolds. If f : M → N is a con-
tinuous map such that for every dierentiable function φ on N the composite φ ◦ f is
dierentiable, then f is dierentiable.
Proof. In fact, for any x ∈ M and any dierentiable function φ in a neighbourhood of
f(x), we have to show that φ ◦ f is dierentiable in a neighbourhood of x. Let ˜φ be a
dierentiable function on N coinciding with φ in a neighbourhood of f(x); then we are
given that ˜φ ◦f is dierentiable. But then ˜φ ◦f and φ ◦f coincide in a neighbourhood
of x, proving our assertion.
3.15. Product manifolds. Let M and N be dierential manifolds. Then one can
provide the topological space M ×N with the structure of a dierential manifold in the
following way. Cover M and N by coordinate charts c
i
: U
i
→ V
i
and c
′
j
: U
′
k
→ V
′
l
.
Then we may cover M × N by U
i
× U
′
k
. On each open set U
i
× U
′
k
one may dene
a coordinate chart c
i
× c
′
k
onto an open set in R
m
× R
n
. The compatibility condition
that needs to be veried, namely the dierentiability of (c
j
× c
′
l
) ◦ (c
i
× c
′
k
)
−1
, follows
obviously from those of c
j
◦c
−1
i
and c
′
l
◦c
′
k
−1
. Thus we have provided the space M ×N
with the structure of a dierential manifold. Now it is easy to check from this denition,
that a mapping of any dierential manifold L into M ×N is dierentiable if and only if
its composites with the projections to M and N are both dierentiable. This manifold
is (therefore) called the product of M and N.
3.16. Submanifolds. We have already dened the notion of an open submanifold.
One may dene a closed submanifold as follows. Let M be a closed subset of a dierential
manifold N with the property that for every m ∈ M , there exists a coordinate chart
(U, c) around m in N such that M ∩ U is the set of common zeros of some of the
coordinates of the chart c.
The rationale of the denition is quite clear. If U is an open submanifold of R
n
, then
the set of points (x) in U satisfying x
1
= x
2
= ··· = x
r
= 0 , should clearly be dened to
be a closed submanifold of U. Note that this set is itself a manifold with the remaining
coordinates serving as a coordinate chart.
Combining the notions of an open submanifold and a closed submanifold, we may
dene a locally closed submanifold, to be a closed submanifold of an open submanifold.
27
Chapter I: Sheaves and Differential Manifolds:
Definitions and Examples
3.17. Immersed manifolds. There is a more general notion of a subset of a manifold
that has the structure of a manifold, which is formally similar to the above notion, but
subtler. Suppose M and N are dierential manifolds. First of all, assume that we have
an injective dierentiable map M → N . Secondly, for any point m ∈ M , we require that
there is a coordinate chart (U, c) in N containing the image of m and a neighbourhood
U
′
of m in M such that U
′
maps into U and its image is the set of common zeros of
some of the coordinates in the chart c. Then we say that M is a manifold immersed in
N.
The notion of an immersed manifold is somewhat delicate for the following reason.
Let us identify M with its image in the following discussion. Notice that we have not
required that for every point m of M, there is a coordinate chart (U, c) of N such that
U ∩M is dened by the vanishing of some of the coordinates of the chart c. The dierence
is not slight! Indeed, the topology of the immersed manifold M is not necessarily that
of its image induced from that of N .
Let us consider an example. We know that the real line R and the torus S
1
× S
1
are both dierential manifolds. Identify S
1
with the submanifold of C consisting of
complex numbers of absolute value 1. Let α ∈ R. Consider the map f : R → S
1
× S
1
given by f (x) = (exp(2πix), exp(2πiαx)). It is clearly dierentiable. It is also injective
if α is irrational. For, if f (x) = f(y), then x − y ∈ Z on the one hand and also,
α(x − y) ∈ Z on the other. Consider the maps g : R → R × R given by x 7→ (x, αx)
and h : R × R → S
1
× S
1
given by (x, y) 7→ (exp(2πix), exp(2πiy)). Clearly we have
f = h ◦g. It is obvious that g imbeds R as a closed submanifold of R ×R. On the other
hand, h is a local isomorphism of dierential manifolds and we conclude that f makes
R an immersed manifold in S
1
× S
1
in our sense. It is easy to see that the image with
the induced topology is not even locally connected. Indeed this shows that the image is
not a subspace at any point.
If α is irrational, it goes round and round innitely. If it is rational, it rewinds at a
nite stage.
We will indicate many simpler examples as well. A gure like 6 (open at the top
28
Section 3: Differential Manifolds
end) can be realized as a submanifold of R
2
by mapping R dierentiably like
Clearly this gure with the topology induced from that of R
2
is not a manifold at
the nodal point.
Again a gure like 8 can be realised as an immersed manifold of R
2
in two dierent
ways, namely by parametrising it as follows.
This example also drives home the point that a closed submanifold is not just an
immersed manifold whose point set is closed. It also needs to have the induced topology.
29