Differential Manifolds

Posted on 2025-06-19 In Math

Outline

Chapter I: Sheaves and Differential Manifolds:

Definitions and Examples

3 Dierential Manifolds

After these preliminaries, we are now ready to dene the concept of dierential man-

ifolds. These objects provide the proper setting for developing dierential and integral

calculus. In Physics, these are called conguration 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. Denition. A dierential 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

= 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

such

that the restriction of A to U is the inverse image of the sheaf of dierentiable functions

on V .

The homeomorphisms referred to above, dened in a neighbourhood of any point,

are called coordinate charts. This is because composing with this homeomorphism the

coordinate functions in R

, one obtains functions x

, . . . , x

. 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 dierential manifolds is based on the notion of dierentiability

in R

, it is clear that R

together with the sheaf of dierentiable functions is a

dierential manifold. Slightly more abstractly, any nite-dimensional vector space

over R is a dierential manifold.

2) If (M, A) is a dierential manifold, and U an open subset of M , then the subspace

U, together with the restriction A|U of A is a dierential 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

(or any

nite-dimensional vector space) is a dierential 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

. Incidentally, it is also a group under matrix multiplication

and is called the General Linear group.

4) Let f be a dierentiable function in R

. Consider the closed subspace (the zero

locus of f) of R

given by

= {x ∈ R

: f(x) = 0}

Section 3: Differential Manifolds

Then Z

has a natural structure of dierential manifold, if at every x ∈ Z

, at least

one of the partial derivatives of f does not vanish. In fact, consider the association

to any open set U ⊂ Z

, of the set of functions on U , which can be extended to

a dierentiable function on a neighbourhood of U in R

. This gives a presheaf

on Z

. (Actually it will turn out that it is a sheaf, but we do not need it here.)

Let A

be the associated sheaf. Then by our assumption, for any x ∈ Z

, there

exists a neighbourhood N of x in R

such that, one of the partial derivatives, say

∂f

∂x

, is nonzero. By the implicit function theorem, the projection to R

n−1

taking

, . . . , x

) to (x

, . . . , x

n−1

) is a dierentiable isomorphism of N ∩ Z

with an

open set V in R

n−1

. (that is to say, a dierentiable bijective map from N onto

V whose inverse is also dierentiable). It is now clear that this isomorphism gives

the local requirement of the sheaf A

. If we further specialise the function to be

f(x) =

∑

− 1, then Z

is the set of vectors of unit length in R

which we call

the unit sphere S

n−1

On the contrary, if f is taken to be the function xy dened in R

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

a dierential manifold in this case. In fact, the topological space Z

cannot have

a structure of a dierential manifold, since it is easy to see that no neighbourhood

of 0 in Z

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

), 1 ≤ i ≤ r, be nitely many dierentiable

functions in R

. Then the closed subspace

= {x ∈ R

: f

(x) = 0 for all i}

has a natural structure of a dierential manifold, if f satises the following con-

dition. For every x ∈ Z

, the rank of the (r, n) matrix

(

∂f

∂x

)

is r. One may think

of f as a function into R

We will now take for f the function on (n, n)-matrices with values also in R

given by A 7→ AA

′

−I

, where A

′

denotes the transpose of A. Then one can check

that the above criterion is fullled and hence the topological space {A ∈ M

(R) :

′

= I

} is a dierential 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

, 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

dened to be the quotient of the unit

Chapter I: Sheaves and Differential Manifolds:

Definitions and Examples

sphere S

by the identication of antipodal points x and −x. We may dene a

sheaf on RP

by associating to any open set U ⊂ RP

, the algebra of dierentiable

functions on its inverse image in S

which are invariant under the antipodal map.

For any point a ∈ S

, the open neighbourhood of a consisting of those y ∈ S

whose distance from a is less than 1 is mapped homeomorphically on an open set

in RP

, and it is easy to see that this takes the sheaf of dierentiable functions

on S

isomorphically onto the sheaf dened above. Thus we see that RP

is a

dierential 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 dierential 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 dierential manifolds. Often, a structure of a dierential mani-

fold is given on a Hausdor topological space M with a countable base for open sets, by

the following procedure. Suppose {U

} is an open covering, and that each U

is provided

with a subsheaf A

of the sheaf of continuous functions making (U

, A

) a dierential

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 dierential manifold. We have already seen 2.5

how we can glue them together. What we need is simply that A

∩U

is the same as

∩U

. In other words, the open submanifold U

∩U

of U

is the same as the open

submanifold U

∩ U

of U

In particular, if (V

, A

) are open submanifolds of R

, then the glueing data may also

be formulated as follows. The space M is covered by open sets U

. For each i, one is

given a homeomorphism c

of U

with the open subset V

of R

. If U

and U

intersect,

then U

∩ U

has as images in V

and V

, two open sets which we may call V

and V

Then c

◦ c

−1

gives a homeomorphism V

→ V

. One may use the homeomorphism

to transport the sheaf of dierentiable functions on V

to a sheaf of algebras on U

But in order to glue these together, we need to know that its restriction to U

∩ U

the same as the restriction of the transported sheaf on U

. This can be achieved if and

only if the above homeomorphism V

→ V

is dierentiable for every i, j. (Note that

the inverse is also dierentiable, by reversing the roles of i and j.) This is in fact the

traditional denition of a dierential manifold.

For example, consider the sphere in R

. The map

(

, . . . , x

)

7→

(

, . . . ,

, . . . , x

)

Section 3: Differential Manifolds

is a homeomorphism of the open set consisting of points of the sphere for which x

> 0

onto the open unit ball in R

n−1

. We can therefore provide these open sets with the

dierential manifold structure of the unit ball. We could do the same with the open

sets of the sphere in which x

< 0. It is clear that all these open sets, as i varies, cover

n−1

. To glue these up, we have only to check that the map

(

, . . . , y

n−1

)

7→

(

, . . . ,

√

1 − |y|

, . . . , ˆy

, . . . , y

n−1

where the term

√

1 − |y|

occurs at the i-th place, is a dierentiable 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

may also be dened as the quotient of R

n+1

\ {0} by

the equivalence relation: (x

, x

, . . . , x

) ∼ (y

, y

, . . . , y

) if and only if there exists a

nonzero real number a such that y

= ax

, for all i. An analogous denition makes sense

over complex numbers as well. In other words, dene the complex projective space CP

to be the quotient of C

n+1

\ {0} by the equivalence relation:

, . . . , z

) ∼ (z

′

, . . . , z

′

)

if and only if there exists a ∈ C

such that z

′

= az

for all i. Since for any such

z = (z

, . . . , z

) at least one z

is nonzero, CP

is covered by open sets which are

images of sets U

, i = 0, . . . , n under the natural map C

n+1

\ {0} → P

, where U

{z = (z

, . . . , z

) : z

6= 0}. It is clear that the subspace z

= 1 of U

is mapped

homeomorphically onto the image of U

in CP

. On the other hand, it is homeomorphic

to C

under the projection which omits the i-th coordinate. This can be used as above

to put a structure of a dierential manifold on it. To check the condition for glueing up,

we only have to check that the map

{(z

, . . . , z

), z

= 1 , z

6= 0 } 7→ {(z

, . . . , z

), z

6= 0 , z

= 1 }

Chapter I: Sheaves and Differential Manifolds:

Definitions and Examples

given by (z

, . . . , z

) 7→ (z

, . . . , z

) is a dierentiable 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 dened 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

of P (V ). What we did above amounts to using this homeomorphism to dene

a dierential structure on X

. We may glue all these structures together to obtain a

dierential 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 dierential manifold. Firstly, let W be any such subspace. Consider the

image of the one-dimensional space Λ

(W ) in Λ

(V ). Thus to every element of the

Grassmannian we have associated an element of the projective space P (Λ

(V )). Then

one checks the following assertion.

3.5. Lemma. If ω is any nonzero element of Λ

(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

(W ) for some r-dimensional subspace W of V .

Proof. In fact, it is obvious that if ω belongs to Λ

(W ), then every element of W is in

the kernel of the above map. Let {e

}, 1 ≤ i ≤ n, be a basis of V such that the rst r of

these generate W . In other words, e

∧ ··· ∧ e

can be taken to be ω. Let v =

∑

;

then v ∧ ω = 0 if and only if a

= 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

,...,i

∧···∧e

we deduce that if e

∧ ω = 0, then a

,...,i

= 0 whenever j does not belong to the set

, . . . , i

}. Since we have assumed that e

∧ ω = 0 for all j ≤ r, we see that the only

nonzero coecient in the expression for ω is a

1,...,r

, that is to say, ω ∈ Λ

(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

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 dierential manifold on the Grassman-

nian is the following. In order to introduce a coordinate system in a neighbourhood of

any r-dimensional subspace U

of V , we proceed as follows. Fix an (n −r)-dimensional

Section 3: Differential Manifolds

subspace W of V supplementary to U

. Consider the set of all r-dimensional subspaces

U of V which are supplementary to W . This set evidently contains U

. The projection

of V onto W corresponding to the direct sum decomposition V = U ⊕ W restricted to

gives a linear map A

of U

into W . When U = U

this map is the zero map. In

general the linear map determines U as the image of η − A

, where η is the inclusion

of U

in V . This sets up a bijection between Hom(U

, W ) and the above set, thereby

coordinatising the Grassmannian in a neighbourhood of U

3.6. Denition. If (M, A) is a dierential manifold, then sections of A over an open

set U of M are called dierentiable functions on U.

Since A is a sheaf, the notion of a dierentiable function on M is of a local nature,

so that the following properties are obvious.

i) If f is a nowhere vanishing dierentiable function, then so is 1/f.

ii) If (U

) is a locally nite covering and (f

) is a family of dierentiable functions

with support in U

, then

∑

is also a dierentiable function.

iii) If φ is a dierentiable function on R

and f

, . . . , f

are dierentiable functions

on M, then φ(f

, . . . , f

) is also a dierentiable function.

3.7. Denition. A continuous map f of a dierential manifold M into another

dierential manifold N is said to be dierentiable if for any x ∈ M , and for every

dierentiable function φ in a neighbourhood U of f(x) in N , the composite φ ◦ f is a

dierentiable function on f

−1

(U).

From the denition of a dierentiable map it follows that there is a homomorphism

of the sheaf A

into f

∗

) and therefore also a homomorphism of A

into f

−1

This is called the structure homomorphism associated to f.

3.8. Remark. It is clear from the denition above that if M, N, P are dierential

manifolds, and f : M → N , g : N → P are dierentiable maps, then the composite

g◦f : M → P is also dierentiable. However, if f : M → N is a dierentiable map which

is bijective, then one cannot conclude that the inverse map is also dierentiable. The

hackneyed counterexample is the function x 7→ x

of R into R, which is dierentiable

and bijective, but whose inverse x 7→ x

is not dierentiable at 0.

3.9. Denition. A dierentiable map f : M → N of dierential manifolds is said to

be a dieomorphism if there is a dierentiable inverse.

A fact, basic to the study of dierentiable functions, is that there are lots of them.

We will now formulate this precisely.

Dierentiable partition of unity.

3.10. Proposition. The following are equivalent.

Chapter I: Sheaves and Differential Manifolds:

Definitions and Examples

i) Given any locally nite open covering (U

)

i∈I

of M, there exist dierentiable func-

tions φ

on M with values in the closed interval [0, 1] such that the support of φ

is contained in U

and

∑

= 1 .

ii) Given open sets U, V with V ⊂ U , there exists a positive dierentiable 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 satises the requirement in ii).

ii) implies i). Let (V

) be a shrinking of (U

), i.e. an open covering with V

⊂ U

for

all i. By Assumption ii), we see that there exist positive dierentiable functions φ

such

that suppφ

⊂ U

and φ

are nonzero everywhere on V

. Now since the covering is locally

nite, the sum φ =

∑

makes sense, is dierentiable and is nonzero everywhere. The

family of functions {ψ

= φ

/φ}, satises 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

such that there is a function as in ii) with V replaced by V ∩ N

and U by U ∩ N

. Then one replaces the sets N

by a locally nite renement of

notes that the sum of the corresponding functions fulls the requirement. Taking N

to be a coordinate neighbourhood of x, we therefore reduce the problem to proving the

following.

3.11. Proposition. Let S

, S

be concentric spheres in R

centered at 0, with S

⊂

. Then there exists a dierentiable function which is nonzero everywhere inside S

and has support contained in the interior of S

Proof. Clearly it is enough to construct a dierentiable function on R

which is nonzero

everywhere inside the unit ball and zero in the complement. The function x 7→ exp

(

∑

−1

)

for all x inside the unit ball and 0 in the complement, is such a function.

3.12. Denition. Let (U

) be a locally nite open covering of a dierential manifold

M. A family (φ

) of dierentiable functions on M with values in [0, 1] is said to be a

partition of unity with respect to the covering (U

), if the support of φ

is contained in

for every i and

∑

= 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 dieren-

tiable function on M. In other words, given a dierentiable function φ in a neighbour-

hood of a closed set K, there exists a dierentiable function ˜φ on M which coincides

with φ in a neighbourhood of K.

Section 3: Differential Manifolds

Proof. In fact, if K ⊂ U , and f is a dierentiable 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 dierentiable 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 dierentiable function on the whole of M as

required.

3.14. Proposition. Let M, N be dierential manifolds. If f : M → N is a con-

tinuous map such that for every dierentiable function φ on N the composite φ ◦ f is

dierentiable, then f is dierentiable.

Proof. In fact, for any x ∈ M and any dierentiable function φ in a neighbourhood of

f(x), we have to show that φ ◦ f is dierentiable in a neighbourhood of x. Let ˜φ be a

dierentiable function on N coinciding with φ in a neighbourhood of f(x); then we are

given that ˜φ ◦f is dierentiable. 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 dierential manifolds. Then one can

provide the topological space M ×N with the structure of a dierential manifold in the

following way. Cover M and N by coordinate charts c

: U

→ V

and c

′

: U

′

→ V

′

Then we may cover M × N by U

× U

′

. On each open set U

× U

′

one may dene

a coordinate chart c

× c

′

onto an open set in R

× R

. The compatibility condition

that needs to be veried, namely the dierentiability of (c

× c

′

) ◦ (c

× c

′

)

−1

, follows

obviously from those of c

◦c

−1

and c

′

◦c

′

−1

. Thus we have provided the space M ×N

with the structure of a dierential manifold. Now it is easy to check from this denition,

that a mapping of any dierential manifold L into M ×N is dierentiable if and only if

its composites with the projections to M and N are both dierentiable. This manifold

is (therefore) called the product of M and N.

3.16. Submanifolds. We have already dened the notion of an open submanifold.

One may dene a closed submanifold as follows. Let M be a closed subset of a dierential

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 denition is quite clear. If U is an open submanifold of R

, then

the set of points (x) in U satisfying x

= x

= ··· = x

= 0 , should clearly be dened 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

dene a locally closed submanifold, to be a closed submanifold of an open submanifold.

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 dierential manifolds. First of all, assume that we have

an injective dierentiable 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

′

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

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 dened by the vanishing of some of the coordinates of the chart c. The dierence

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

× S

are both dierential manifolds. Identify S

with the submanifold of C consisting of

complex numbers of absolute value 1. Let α ∈ R. Consider the map f : R → S

× S

given by f (x) = (exp(2πix), exp(2πiαx)). It is clearly dierentiable. 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

× S

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 dierential manifolds and we conclude that f makes

R an immersed manifold in S

× S

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 innitely. 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

Section 3: Differential Manifolds

end) can be realized as a submanifold of R

by mapping R dierentiably like

Clearly this gure with the topology induced from that of R

is not a manifold at

the nodal point.

Again a gure like 8 can be realised as an immersed manifold of R

in two dierent

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.