AN INTRODUCTION TO DIFFERENTIAL GEOMETRY

DIFFERENTIAL GEOMETRY

Luca Chiantini

January 11, 2021

1 Differential Manifolds 1

1.1 Differential structures . . . . 2

1.2 Equivalent differential structures . . . . 10

1.3 Differential maps . . . . 13

1.4 Exercises . . . . 17

2 Tangent vectors 19 2.1 The tangent bundle . . . . 20

2.2 Derivation on manifolds . . . . 27

2.3 Vector fields . . . . 34

2.4 Frames . . . . 38

2.5 Exercises . . . . 47

3 Generalities on vector bundles 49 3.1 Transition data . . . . 50

3.2 Linear maps . . . . 60

3.3 Sections and frames . . . . 66

3.4 Pull back and push forward . . . . 71

3.5 Exercises . . . . 76

4 Curves and surfaces 77 4.1 Curves . . . . 78

4.2 Arcs in R

. . . . 79

3

4.3 Surfaces, and Caps in R

. . . . 96 4.4 Exercises . . . 107

5 Orientation of manifolds 109

5.1 Orientation in a vector space . . . 110

5.2 Orientation on manifolds . . . 113

5.3 Exercises . . . 120

Differential Manifolds

1

1.1 Differential structures

The initial chapter contains the basic definitions of differential manifold and differential map, together with their first properties and examples.

Definition 1.1.1. Let X be a topological space, which satisfies the separa- tion axiom T

(Hausdorff space). A chart in X is a pair (U, φ) where:

i) U is a connected open subset of X;

ii) φ is a homeomorphism between X and an open (connected) subset φ(U ) of R

.

The number n is the dimension of the chart.

Definition 1.1.2. A collection A = {(U

, φ

)} of charts in X is an atlas (of dimension n) on X if:

i) all the charts have dimension n;

ii) the open sets U

’s cover X, i.e. X = S U

.

In a general atlas the various charts have no interactions, even when they overlap. Differential Geometry only considers atlases in which the passage from one to another chart is realized by maps of class C

.

Definition 1.1.3. A differential structure on the Hausdorff space X is an atlas A = {(U

, φ

)} such that the following condition holds: for all i, j the map:

φ

◦ φ

: φ

(U

∩ U

) → φ

(U

∩ U

) is of class C

.

When A is a differential structure on X, the pair (X.A) is a differential

manifold.

By abuse of notation, we will use expressions like P ∈ (U, φ), i.e. P belongs to the chart (U, φ), or V is a subset of the chart (U, φ) respectively, to indicate that P ∈ U , respectively V ⊂ U .

Similarly, when M = (X, A) is a differential manifold, we will use expres- sions like P is a point of M , to indicate that P is a point of the Hausdorff space X.

Let us observe that we do not exclude the case in which several charts of a differential structure A are defined on the same open subset U . Similarly, we can have several different differential structures on the same Hausdorff space X.

In order to understand the condition that φ

◦ φ

is of class C

, notice that, for all P ∈ (U, φ), one can define the local coordinates of P in (U

, φ

) as the coordinates (x

, . . . , x

) of φ

(P ) ∈ U

⊂ R

. If (U

, φ

) is another chart containing P , and (y

, . . . , y

) are the coordinates of φ

(P ), then:

(y

, . . . , y

) = (φ

◦ φ

)(x

, . . . , x

).

Thus, the functions φ

◦ φ

are the transition function from the local coordinates in one chart to the local coordinates in another chart.

In a differential structure, the transition from local coordinates in one

chart to local coordinates in another chart is mediated by C

functions.

Example 1.1.4. The simplest example of differential structure is the atlas on R

given by a unique chart A = {(R

, id)}.This is called from now on the canonical differential structure on R

.

Example 1.1.5. Let A = {(U

, φ

)} be a differential structure on a topo- logical space X and let U be an open subset of X. Then U has the induced differential structure A

defined by

A

= {(U

, φ

)},

where U

ranges among the connected components of U ∩ U

, for all i.

Example 1.1.6. Let X be a Hausdorff space with a homeomorphism φ : X → U , where U is an open subset of R

. It is clear that if X is connected then {(X, φ)} is a differential structure on X. If X is not connected, one gets a differential structure on X by considering the atlas {(U

, φ

)}, where the U

’s are the connected components of X.

In particular, let f be a continuous, R

-valued function defined over an open subet Ω ⊂ R

. The graph Γ ⊂ R

,

Γ

= {(x

, . . . , x

, f

(x

, . . . , x

), . . . , f

(x

, . . . , x

)) : (x

, . . . , x

) ∈ Ω}

has a canonical differential structure defined by the projection to the first n

coordinates, which sends homeomorphically Γ to Ω.

Example 1.1.7. let S

be the unit circle, defined as the set of points in R

whose coordinates satisfy the equation x

+ y

− 1 = 0. S

is a circle of radius 1, endowed with the topology induced by the canonical topology of R

, thus it is a connected Hausdorff space. We define directly a differential structure of dimension 1 on on S

.

Set P = (0, 1), P

= (0, −1), L = tangent line to S

at P , L

= tangent line to S

at P

. We define on S

a differential structure A with two charts, as follows. The first chart is (U, φ), where U = S

\ {P } and φ is the stereographic projection from P to L

, which is identified with R (see the picture below). The second chart is (U

, φ

), where U

= S

\ {P

} and φ

is the stereographic projection from P

to L.

By simple geometric arguments, it follows that for any point H ∈ U ∩U

= S

\ {P, P

}, the triangles P, φ(H), P

and P, φ

(H), P

are similar. So one computes that:

P φ

(H)

P P

= P P

φ(H)P

.

Since the segment P P

is a diameter, of length 2, then for any point x = φ(H) ∈ φ(U ∩ U

) one computes that

(φ

◦ φ

)(x) = φ

(H) = φ

(H)P = 4

x .

Since φ(U ∩ U

) corresponds to R − {0}, it follows that the function (φ

◦ φ

)(x) = 4/x is of class C

on φ(U ∩U

). A perfectly analogue computation shows that (φ ◦ φ

)(y) = 4/y is of class C

on φ

(U ∩ U

) = R − {0}. It follows that A = {(U, φ), (U

, φ

)} is a differential structure on S

.

We will refer to the differential structure on S

defined in Example 1.1.7 as the stereographic structure.

With a similar construction, by using stereographic projections in higher dimensions, one can define a stereographic structure of the n-dimensional sphere S

(of radius 1), defined as the set of points in R

which satisfy x

+ · · · + x

− 1 = 0.

Example 1.1.8. Let X be the subset of R

defined by an implicit equation

F (x

, . . . , x

) = 0

where F : R

→ R is a function of class C

.

If the partial derivatives ∂F/∂x

do not vanish simultaneously at any point P ∈ X, then one can define a differential structure on X as follows.

By the Implicit Function Theorem (??) there exists an open cover {U

} of X such that each U

is the graph of a C

function f

: Ω

→ R, Ω

being an open subset of R

. After refining the cover, we may assume that each U

is connected. Thus, on any U

one can define a chart as in the second part of example 1.1.6, and these charts all together define an atlas A (of dimension n) on X.

Indeed, A is a differential structure. Namely, on U

∩ U

the transition

function φ

◦ φ

is given as follows. Assume that in a point of U

the

derivative ∂F/∂x

is non-zero. Then U

coincides with a set of points of

type (x

, . . . , x

, f

(x

, . . . , x

)). Similarly, if in a point of U

the deriva-

tive ∂F/∂x

is non-zero, then U

coincides with a set of points of type

(f

(x

, . . . , x

), x

, . . . , x

). Thus for points (x

, . . . , x

) in φ

(U

∩ U

) φ

◦ φ

(x

, . . . , x

) = φ

((x

, . . . , x

, f

(x

, . . . , x

)) =

= (x

, . . . , x

, f

(x

, . . . , x

)), which is clearly of class C

, because f

is.

One can easily generalize the example to spaces X implicitly defined by the simultaneous vanishing of several C

functions F

= · · · = F

= 0, provided that in any point of X the jacobian matrix (∂F

/∂x

) has (maximal) rank min{n + 1, m}.

We will refer to the previous differential structure as the implicit differen- tial structure on X.

When, in the previous example, all the functions F

, . . . , F

are polyno- mials, then X is called a smooth algebraic variety. Thus, all smooth algebraic varieties have an implicit differential structure.

Example 1.1.9. The subset of R

implicitly defined by the vanishing of the function F = x

− y

is composed by two lines which cross each other at the origin O. It is an easy topological exercise to prove that the point O has no neighborhoods in X which are homeomorphic to some open subset of R

. Thus it is impossible to define a differential structure (and even an atlas) on X.

Notice that Example 1.1.8 does not apply to X, because in O both partial derivatives ∂F/∂x and ∂F/∂y vanish.

Example 1.1.10. Consider the map φ : R → R given by φ(x) = √

x. The atlas (with one chart) {(R, φ)} is a differential structure on R, because φ is a homeomorphism.

On the other hand, the atlas with two charts {(R, id), (R, φ)} is not a differential structure, because (φ ◦ id

)(x) = φ(x) = √

x is not of class C

at the origin (even if (id ◦ φ

)(y) = y

actually is of class C

at all points).

Example 1.1.11. The Moebius strip is a mathematical object with several amazing properties. It is usually constructed from a rectangular sheet of paper, in which two opposite sides are glued together by reversing the order.

A precise mathematical definition of the Moebius strip reveals that it can be endowed with a natural differential structure.

We will use the following definition for the Moebius strip. Start with the square ¯ Q:

Q = {(x, y) : −1 ≤ x ≤ 1, −1 < y < −1}. ¯

Notice that the two vertical sides belong to ¯ Q, while the horizontal sides do not. Define in ¯ Q the equivalence relation ∼ which, besides reflexivity, associates point (1, y) ∼ (−1, −y) (and symmetrize). The Moebius strip is the quotient M = ¯ Q/ ∼. Notice indeed that ∼ identifies the points (1, y) and (−1, −y), leaving unchanged the rest of ¯ Q. hence M is obtained from ¯ Q by glueing together the vertical sides of the square, after reversing the order, because of the minus sign in front of y.

One can define a differential structure A of dimension 2 on M as follows.

A has two charts (U

, φ

) and (U

, φ

), where:

U

is the open square Q = {(x, y) : −1 < x < 1, −1 < y < −1}, on which the equivalence relation has no effect,

φ

is the identity,

U

is M minus the set of (equivalence classes of) points in the central vertical segment S = {(0, y)} ⊂ M . φ

is defined by:

φ

(x, y) =

( (x − 1, y) if x > 0 (x + 1, −y) if x < 0 .

Notice that φ

is compatible with the equivalence relation, because (1, y) and (−1, −y), which define the same class in ∼, hence the same point in M , are both sent to (0, y).

The image of φ

is again the open square Q and φ

is invertible, with:

φ

(x, y) =

( (x + 1, y) if x ≤ 0

(x − 1, −y) if x ≥ 0

(there is no conflict for x = 0, since (0, y) maps either to (1, y) or to (−1, −y), but the two points coincide in the quotient). The atlas A = {(U

, φ

), (U

, φ

)} is a differential structure because both φ

◦ φ

= φ

and φ

◦ φ

= φ

are defined in the set Q \ S, which is disconnected, and are given by linear, hence C

, maps on the two open subsets of the disconnection.

Example 1.1.12. Extending the construction of Example 1.1.11 (with more

charts) one can define a natural differential structure on the following Haus-

dorff spaces (denoted schematically):

Example 1.1.13. Let M = (X, A), M

= (X

, A

) be two differential man- ifolds. The cartesian product X × X

(which is still a Hausdorff space) can be endowed with a differential structure (product structure) A

defined as follows: if A = {(U

, φ

)} and A

= {(U

, φ

)} then A

= {(U

× U

, φ

× φ

)}.

We will denote this differential manifold by M × M

.

1.2 Equivalent differential structures

In principle, different differential structures A

.A

on the same Hausdorff

space X define different differential manifolds. On the other hand, if charts

of the two differential structures have C

transition functions, then the dif-

ference between (X, A) and (X, A

) is purely formal, because their behavior with respect to Differential Geometry is identical.

Definition 1.2.1. Two differential structures A

.A

on the same Hausdorff space X are equivalent if A ∪ A

is again a differential structure on X. In this case, we also say that the differential manifolds (X, A

) and (X, A

) are equivalent.

The previous definition determines an equivalence relation in the set of all differential structures on X.

Namely, the relation is clearly reflexive and symmetric. To prove transitiv- ity, one needs to use the fact that being of class C

, for a function, is a local property. Take differential structures such that A = {(U

, φ

)} is equivalent to A

= {(U

, φ

)} and A

is equivalent to A

= {(U

, φ

)}. We want to prove that A ∪ A

is a differential structure, which in practice reduces to prove that all transition functions φ

◦ φ

, and their inverse, are C

in their domains.

The domain of φ

◦ φ

is φ

(U

∩ U

). For any point P ∈ U ∩ U

there is a chart (U

, φ

) of A

containing P . In the open neighborhood φ

(U

∩ U

∩ U

) of φ(P ) one can write:

φ

◦ φ

= φ

◦ φ

◦ φ

◦ φ

= (φ

◦ φ

) ◦ (φ

◦ φ

),

which is C

, since the composition of of C

maps is C

. The transitivity follows.

An immediate consequence of the definition is that if A, A

are differential structures on X, and A ⊂ A

, then A, A

are equivalent.

Theorem 1.2.2. Every differential structure A is contained in a unique maximal differential structure.

Proof. To prove the uniqueness, consider two maximal differential structures A

, A

containing A, Since A ⊂ A

, then A, A

are equivalent. Similarly

1

Of course, just saying that elements of a set are equivalent is not sufficient to define

an equivalence relation!

A, A

are equivalent. By transitivity, it follows that A

, A

are equivalent.

Then A

∪ A

is a differential structure. Since, A

, A

are maximal, it follows A

= A

∪ A

= A

, hence uniqueness holds.

To prove the existence, we will apply Zorn’s Lemma to the set of differ- ential structures containing A. The set is partially ordered by inclusion. Let A

⊂ A

⊂ · · · ⊂ A

⊂ . . . is an ascending chain of differential structures containing A. We claim that A

= S A

is a differential structure containing A. Namely, if (U, φ), (U

, φ

) are charts in A

, then there are indices i, j such that (U, φ) ∈ A

and (U

, φ

) ∈ A

. If, say, i ≤ j then both (U, φ), (U

, φ

) belong to A

, which is a differential structure. Thus the transition functions φ

◦ φ

and φ ◦ φ

are both C

. The claim follows.

In some book of Differential Geometry, maximality is included in the very definition of differential structure. We prefer to maintain our definition be- cause in concrete examples it is easier to deal with non maximal structures, with finitely many charts (it is almost immediate to see that every positive dimensional maximal differential structure contains infinitely many charts).

The difference is purely formal, since every differential structure uniquely determines a maximal differential structure.

Example 1.2.3. The circle S

has the stereographic differential structure with two charts, introduced in Example 1.1.7. On the other hand, one can define a differential structure on the circle x

+y

−1 = 0 also via the Implicit Function Theorem of Example 1.1.8. Let us prove that the two structures are equivalent.

Consider for instance the chart (U, φ) of the stereographic structure, where U is the circle minus the point P = (0, 1) and φ is the stereographic pro- jection from P to the line y = −1. Since U

= S

∩ {y > 0} is the graph of the function f (x) = √

1 − x

, then we have a chart (U

, φ

) of the struc- ture defined by the Implicit Function Theorem, where φ

is the orthogonal projection to the x axis.Then φ

(U ∩ U

) is the open interval (−1, 1). For any x ∈ (−1, 1), φ

(x) is the point Q = (x, √

1 − x

). The projection φ(Q)

corresponds to the first coordinate of the intersection of the line P Q with the line y = −1. After some computations of elementary Analytic Geometry, one finds that

φ ◦ φ

(x) = φ(Q) = 2x 1 − √

1 − x

. The function is C

in the domain (−1, 1).

A similar computation proves that even all the remaining transition func- tions are C

.

Example 1.2.4. The canonical maximal differential structure on the space R

is the maximal differential structure equivalent to {(R

, id)}.

The structure {(R, φ)} on R defined in Example 1.1.10, where φ is the map φ(x) = √

x, is not equivalent to the maximal differential structure.

1.3 Differential maps

After introducing the notion of differential manifolds, next step is the defini- tion of maps M → M

between differential manifolds which are compatible with their geometric structure.

Definition 1.3.1. Let M = (X, {(U

, φ

)}) and M

= (X

, {(U

, φ

)}) be differential manifolds. A continuous map f : X → X

defines a differential map between M and M

(which, by abuse, we will denote with f : M → M

) if for all i, j the map

φ

◦ f ◦ φ

: φ

(U

∩ f

(U

))toφ

(U

) is of class C

.

A differential map f : M → M

is a diffeomorphism if there exists a

differential map g : M

→ M such that both f ◦ g and g ◦ f are the identities

of the respective Hausdorff spaces. Two differential manifolds M, M

are

diffeomorphic if there exists a diffeomorphism f : M → M

.

Example 1.3.2. Consider the differential manifolds R

and R

, with the canonical structure. A map R

→ R

is a differential map between the two manifolds if and only if it is of class C

.

Example 1.3.3. An algebraic map between two smooth algebraic variety is locally defined by quotients of polynomials. Thus, any such map is a dif- ferential map of the corresponding differential manifolds (with the structure defined by the Implicit Function Theorem).

Example 1.3.4. The map f : R → R which sends x to x

is a differential function which is not a diffeomorphism, because the inverse g(y) = √

y is not of class C

at the origin.

The following properties of differential maps are elementary and the proof

is left as an exercise.

Proposition 1.3.5. a) Let M be a differential manifold on the Hausdorff space X. Then the identity X → X defines a differential map M → M (notice that we use the same differential structure, see Proposition 1.3.7).

b) The composition of differential maps is a differential map.

c) A constant map is a differential map.

d) If f, g : M → R

are differential maps, then also f + g : M → R

is a differential map.

e) If f, g : M → R are differential maps, then also f g : M → R is a differential map.

f ) If f : M → M

and g : N → N

are differential maps, then also f × g : M × N → M

× N

is a differential map.

From Proposition 1.3.5 d), it follows that the set of all differential maps M → R

is a linear space. From Proposition 1.3.5 e), the set of differential maps from M to R is a ring, which we will denote by C

(M ).

Since being of class C

is a local property, the following proposition holds.

Proposition 1.3.6. let M = (X, A) and N = (Y, B) be two differential manifolds. A continuous map f : X → Y defines a differential map M → N if and only if for every open subset U of X the restriction f

: U → Y defines a differential map U → N , where we consider on U the differential structure induced by A.

Next proposition clarifies the connection between differential maps and equivalent differential structures.

Proposition 1.3.7. Let M = (X, A) and N = (Y, B) be differential mani-

folds. Let A

be a differential structure on X equivalent to A, and let B

be

a differential structure on Y equivalent to B. A map f : X → Y defines a

differential map M → N if and only if the same map defines a differential map from M

= (X, A

) to N

= (Y, B

).

Proof. Assume that f : M → N is a differential map. consider a chart (U

, φ

) of A

and a chart (V

, ψ

) of B

. We need to prove that ψ

◦ f ◦ φ

is C

. It is sufficient to prove that for any P ∈ U

∩ f

V

, the map ψ

◦ f ◦ φ

is C

in a neighborhood of φ

(P ). Consider a chart (U, φ) of A containing P and a chart (V, ψ) of B containing f (P ). In the open neighborhood φ

(U ∩ U

∩ f

(V ) ∩ f

(V

)) one has:

ψ

◦ f ◦ φ

= (ψ

◦ ψ

) ◦ (ψ ◦ f ◦ φ

) ◦ (φ ◦ φ

).

The map ψ ◦ f ◦ φ

is C

because f : M → N is a differential map. The maps ψ

◦ ψ

and φ ◦ φ

are C

because A is equivalent to A

and B is equivalent to B

. Thus f : M

→ N

is a differential map.

The converse holds for the same reason.

Corollary 1.3.8. Equivalent differential structures A, A

on X define diffeo- morphic manifolds M = (X, A) and M

= (X, A

).

We finish with an important examples of differential manifolds which mix the differential structure with a group straucture.

Definition 1.3.9. Let X be a Hausdorff topological group and let A be a differential structure on X. The differential manifold G = (X, A) is a Lie group if the multiplication defines a differential map G × G → G and the inverse defines a differential map G → G.

Example 1.3.10. The space R

od all square matrices of type a × a can be

identified with R

. let X be the open subset of R

formed by matrices with

non zero determinant, i.e. invertible matrices. The row by column product

defines on X the structure of topological group. As the product of matrices

is defined in terms of sums and products of the entries, which are local

coordinates in the canonical structure of R

, both the maps X × X → X

defines by the product and the mapX → X defined by the inverse are of class C

. It follows that X, with the structure induced by the canonical structure on R

, is a Lie group. We will denote this group by GL(a).

Example 1.3.11. The circle S

is a Lie group. Indeed one can identify S

with R/2πZ, a group in which the sum of points is defined by the sum of the corresponding angles. The sum of angles is clearly a C

operation on S

.

1.4 Exercises

Exercise 1.4.1. Show that if X is a compact topological space, then every differential structure on X is equivalent to a differential structure with a finite number of charts.

Exercise 1.4.2. Prove that every maximal differential structure on R has infinitely many charts.

Exercise 1.4.3. Give a proof of Proposition 1.3.5.

Exercise 1.4.4. Prove that if M = (X.A) and N are differential manifolds and f : M → N is a differential map, then there exists a refinement A

of A such that for all charts (U

, φ

) of A

the image f (U

) is contained in some chart of N .

Exercise 1.4.5. Prove that C

(M ) is a vector space over R.

Exercise 1.4.6. Prove that if G, G

are Lie groups, then also G × G

is a Lie

group.

Tangent vectors

19

2.1 The tangent bundle

We introduce here the notion of tangent vectors to a differential manifold, and the tangent bundle.

Tangent vectors applied in a point P of a differential manifold M describe how the manifold spreads in a neighborhood of P .Tangent vectors are our main tool to explore the shape of a differential manifold and its analytic properties (derivatives of functions).

We proceed as we will do continuously in the book: first we define the notion of tangent vector at a point of R

, then we extend the definition to points in a general differential manifold.

Definition 2.1.1. let P be a point in R

. The space T

of tangent vectors to R

at P is simply the space ov vectors of R

applied in P .

For every C

function f defined in an open subset Ω ⊂ R

to R

con- taining P , the differential df defines a linear function df

: T

→ T

. The differential is the map associated, with respect to the canonical basis of R

, R

to the jacobian matrix of f , i.e. the matrix whose entries are the partial derivatives of f .

Let M be a differential manifold of dimension n and let P be point of M , belonging to the chart (U, φ). Then, one can use the homeomorphism between U and the open subset φ(U ) of R

to define angent vectors to M at P as vectors of R

applied in φ(P ). Before doing that, a problem must be solved: if P belongs to two charts (U, φ) and (U

, φ

), one need to analyze the relations between vectors of R

applied in φ(P ) and vectors of R

applied in φ

(P ).

Proposition 2.1.2. Let P be a point of the differential manifold M belong- ing to the charts (U, φ) and (U

, φ

). Then the differential of the transition function d(φ

◦ φ

), computed in φ(P ), defines an isomorphism of vector spaces

d(φ

◦ φ

)

: T

→ T

.

Proof. The map φ

◦ φ

is invertible around φ(P ), the inverse being φ ◦ φ

. By the Inverse Function Theorem, also the differential d(φ

◦ φ

)

is invertible, the inverse being d(φ◦φ

)

, because φ

(P ) = (φ

◦φ

)(φ(P )).

In our definition of tangent vectors, we will use the isomorphism d(φ

◦ φ

)

to identify vectors of R

applied in φ(P ) with vectors of R

applied in φ

(P ).

We are now able to define tangent vectors to M at a point P belonging to the chart (U, φ) as vectors of R

applied in φ(P ). If P belongs to another chart (U

, φ

), we will use the isomorphism d(φ

◦ φ

)

to match the two definitions.

Technically, we proceed as follows.

Let M = (X, {(U

, φ

)}) be a differential manifold. Consider the disjoint union T = t(U

× R

), which is naturally a Hausdorff topological space.

Elements of T are pairs (P, v)

where P is a point of U , v is a vector of R

, and the pair is indexed by the chart (U, φ).

We introduce in T the following relation:

(P, v

)

∼ (Q, v

)

if and only if P = Q and v

= d(φ

◦ φ

)

(v

).

It is easy to see that the relation is indeed an equivalence relation. Namely:

ˆ φ

◦ φ

(same index) is the identity in a neighborhood of φ

(P ), thus also its differential is the identity, hence the relation is reflexive.

ˆ As observed above, d(φ

◦ φ

)

is the inverse of d(φ

◦ φ

)

, thus if v

= d(φ

◦ φ

)

(v

), then also v

= d(φ

◦ φ

)

(v

), and the relation is symmetric.

ˆ If (U

, φ

), (U

, φ

), (U

, φ

) are charts containing P , with (P, v

)

∼

(P, v

)

and (P, v

)

∼ (P, v

)

, then v

= d(φ

◦φ

)

(v

)

and v

= d(φ

◦ φ

)

(v

), so that:

v

= (d(φ

◦ φ

)

◦ d(φ

◦ φ

)

)(v

) =

= d(φ

◦ φ

◦ φ

◦ φ

)

)(v

) = d(φ

◦ φ

)

(v

) and also transitivity holds.

Definition 2.1.3. The quotient T

= T / ∼ is the tangent bundle to the differential manifold M . Its elements are the tangent vectors to M . We will denote by χ the natural map T → T

.

Notice that a tangent vector τ ∈ T

is an equivalence class τ = [(P, v

)

].

In any pair of the equivalence class the point P is the same. Thus it depends only on τ . P is denoted as the application point of the tangent vector τ .

On the other hand, the second element of the pair v

depends on the chart (U

, φ

). Notice that the elements of the equivalence class τ are in one-to- one correspondence with the charts of M containing the application point P . Namely, if (U

, φ

) contains P , then the pair (P, v

)

) belongs to the requivalence class, where v

= d(φ

◦ φ

)

(v

). Moreover, two pairs (P, v)

and (P, w)

indexed by the same chart cannot belong to the same equivalence class unless w = d(φ

◦ φ

)

(v) = v, i.e. the two pairs are equal. Thus, for any chart (U

, φ

) containing the application point P , τ has exactly one representative (P, v) indexed by the chart. The second element v of the pair, which is a vector in R

, is called the vector part of τ in the chart (U

, φ

).

Summarizing, a tangent vector τ to M is defined by:

ˆ an application point P , which is uniquely determined;

ˆ a vector part v ∈ R

, which depends on the choice of a chart containing P . Vector parts v

, v

of τ relative respectively to the charts (U

, φ

) and (U

, φ

), containing P , are related by:

v

= d(φ

◦ φ

)

(v

).

Definition 2.1.4. The set of all tangent vectors applied in a point P ∈ X is the tangent space to X at P and it is denoted by T

.

Clearly, the choice of a chart (U, φ) containing P determines a bijection T

→ R

, by sending τ ∈ T

to its vector part in the chart (U, φ).

We prove that the bijection introduces in T

a natural structure of vector space.

Proposition 2.1.5. For τ

, τ

∈ T

and a ∈ R define the sum τ

+ τ

and the product aτ

as follows: choose a chart (U, φ) containing P and let v

, v

be the vector parts of τ

, τ

respectively, in the chart (U, φ). Then

ˆ τ

+ τ

is the tangent vector applied in P , whose vector part in (U.φ) is v

+ v

;

ˆ aτ

is the tangent vector applied in P , whose vector part in (U.φ) is av

.

We obtain in this way operations in T

which determine the structure of vector space of dimension n = dim(M ).

Proof. The only non-trivial part of the statement is to prove that the defini- tion of the operations does not depend on the choice of the chart (U, φ). If (U

, φ

) is another chart containing P , then the vector parts of τ

, τ

in the new chart are d(φ

◦ φ

)

(v

) and d(φ

◦ φ

)

(v

) respectively. Since the differential is linear, we obtain the equalities:

d(φ

◦ φ

)

(v

) + d(φ

◦ φ

)

(v

) = d(φ

◦ φ

)

(v

+ v

) d(φ

◦ φ

)

(av

) = a d(φ

◦ φ

)

(v

),

which prove the claim.

We can see now the reason for the name tangent bundle: T

can be considered as the union of many vector spaces of dimension n = dim(M ), parameterized by the points of M .

Example 2.1.6. Let S

be the unit circle, with the differential structure {(S

\ {P }, φ), (S

\ {Q}, φ

)}, defined in Example 1.1.7. The tangent bundle T

is thus the quotient of the disjoint union

(S

\ {P } × R) t (S

\ {Q} × R)

through the equivalence relation which identifies the pairs (Q, v)

and (Q, v

)

when v

= −v/z

, where z = φ(Q) ∈ R.

For instance, if Q = (1/ √ 2, 1/ √

2), then the tangent vector applied in Q with vector part v in the first chart (S

\{P }, φ) has vector part (2 √

2−3)v/4 in the second chart.

One of the basic points in Differential Geometry is the fact that T

has a natural differential structure, whose charts are in one-to-one correspondence with the charts of M . We define the structure with a series of remarks.

ˆ We can define over T

the quotient topology obtained from the product topology on the disjoint union T . Since M is Hausdorff, then it turns out that also T

becomes a Hausdorff space.

ˆ There exists a canonical surjective map π : T

→ M (projection)

which sends a tangent vector τ to its application point π(τ ) (which

is uniquely defined). The map π is continuous. Indeed, if U is open

in M , then π

(U ) (which corresponds to the set of tangent vector

whose application point sits in U ) has an inverse image in the quotient

χ : T → T

which is equal to the disjoint union t(U ∩ U

) × R

, where

U

ranges over the open sets which define the charts of M . Thus π

(U )

is always open.

ˆ For any chart (U

, φ

) of M there is a natural map ν

: π

(U

) → R

which sends τ ∈ π

(U

) to its vector part in the chart (U

, φ

).

ˆ By the construction of the topology on T

, the map ¯ ν

: π

(U

) → U

× R

which sends τ to (π(τ ), ν

(τ )) is a homeomorphism.

ˆ For any chart (U

, φ

) of M we define a chart (of dimension 2n) on T

by taking (π

(U

), Φ

), where Φ

= (φ

◦ π) × ν

.

In order to prove that {(π

(U

), Φ

)} is a differential structure on T

we need the following:

Lemma 2.1.7. Let Ω ⊂ R

be an open subset and let f : R

→ R

be a function of class C

. Call J

the jacobian matrix of f in the point (x

, . . . , x

) ∈ Ω. Then the function f

: Ω × R

→ R

defined by

f

(x

, . . . , x

, v

, . . . , v

) = (f (x

, . . . , x

), J

(v

, . . . , v

)) is again of class C

.

Proof. The first n components of f

coincide with those of f , thus they are C

. The remaining m components are given by the product of the matrix J

(whose entries are the derivatives of f , hence they are C

functions of the x

’s) and the column (v

, . . . , v

). The result of the product is clearly C

in all the variables x

, . . . , x

, v

, . . . , v

.

Proposition 2.1.8. The atlas ¯ A = {(π

(U

), Φ

)} is a differential structure of dimension 2n on T

.

The map π : T

→ M defined above, which sends every tangent vector to its application point, is a differential surjective map.

Proof. We know by definition that ¯ A is an atlas of dimension 2n on T

. For all i, j the transition function Φ

◦ Φ

corresponds to

Φ

◦ Φ

= (φ

◦ π) × ν

◦ (φ

◦ π

) × ν

= (φ

◦ φ

) × (ν

◦ ν

).

The function ν

◦ ν

sends (v

, . . . , v

) to the vector part in (U

, φ

) of the tangent vector whose vector part in (U

, φ

) is (v

, . . . , v

). Thus ν

◦ ν

corresponds to d(φ

◦φ

)

, which is associated to the jacobian matrix.

Thus we may apply Lemma 2.1.7 to conclude that Φ

◦ Φ

is of class C

. To prove the second claim, take any charts {(π

(U

), Φ

)} of T

and (U

, φ

) of M . Then one computes:

(φ

◦ π ◦ Φ

)(x

, . . . , x

, v

, . . . , v

) = (φ

◦ φ

)(x

, . . . , x

),

thus the map (φ

◦ π ◦ Φ

) is clearly of class C

.

Observe that the charts of ¯ A are in one-to-one correspondence with the charts of M .

From now on, when we take the tangent bundle of a differential mani- fold M , we will always consider it as a differential manifold itself, with the structure ¯ A defined in the previous proposition.

Notice that the tangent bundle to M can be seen as the (disjoint) union of many linear spaces. Indeed, as observed above, T

is the union of the spaces

T

= {the set of tangent vectors applied in P }.

Thus, it turns out that the tangent bundle of any differential manifold has a double nature. From one side it is a differential variety, thus an object of Differential Geometry. On the other hand, it can be explored also with algebraic tools, because, as a union of linear spaces, it is strictly related to Linear Algebra.

The double nature of the vector bundle (which will be generalized in the

definition of general vector bundles, see chapter 3) will be a source of impor-

tant methods for the study of differential manifolds.

2.2 Derivation on manifolds

Tangent vectors are indeed the objects that define in which direction one can move, starting from P , and staying in the differential manifold M . In terms of tangent vector, one can define the directional derivative of a real-valued function.

Definition 2.2.1. let M be a differential manifold and let f : M → R be a differential map (as usual, we consider R as a differential manifold with the structure defined in Example 1.1.4). For any tangent vector τ ∈ T

we define the (directional) derivative ∂f /∂τ of f along τ as follows. Choose a chart (U, φ) of M which contains the application point P = π(τ ) of τ . Then define:

∂f

∂τ = ∂(f ◦ φ

)

∂v = grad(f ◦ φ

)

· v, where v is the vector part of τ in (U, φ).

Apparently, the previous definition depends on the choice of a chart con- taining the application point P of τ . We show that this is not the case.

Namely, if (U

, φ

) is another chart containing P , and v

is the vector part

of τ in the new chart, then v

= d(φ

◦ φ

)

(v). Since the gradient of a

function from R

to R is exactly its differential, then:

grad(f ◦ φ

)

· v

= d(f ◦ φ

)

(d(φ

◦ φ

)

(v)) =

= d(f ◦ φ

◦ φ

◦ φ

)

(v) = grad(f ◦ φ

)

· v.

Example 2.2.2. Consider the circle S

, with the differential structure de- fined in Example 1.1.7. Consider the map f : S

→ R which sends the point (x, y) to x and consider the tangent vector τ , applied in the point B = (1/ √

2, 1/ √

2) of the circle, whose vector part v in the chart (S

\{P }, φ

) is 2. Let us compute the derivative of f along τ .

The map φ

(stereographic projection from (0, 1), sends B to φ

B) = 2.

For any t ∈ R, one computes, by analytic geometry, that φ

(t) is the point of S

of coordinates (4t/(t

+ 4), (t

− 4)/(t

+ 4)). Thus f ◦ φ

sends t to 4t/(t

+ 4). Then one computes:

∂f

∂τ = ∂(f ◦ φ

)

∂v = grad(f ◦ φ

)

· v =

=

 d(

) dt



{2}

· 2 =  4t

− 8t + 16 (t

+ 4)



{2}

· 2 = 1

4 · 2 = 1 2 . The following proposition is straightforward and the proof is left as an exercise.

Proposition 2.2.3. let f, g be differential functions from M to R and let a be any scalar. Then for any tangent vector τ

(a)

=

+

, (b)

= g

+ f

, (c)

= a

.

With a similar trick, for any differential map f : M → N between dif-

ferential manifolds, we can define the differential df of f between the two

tangent bundles df : T

→ T

.

Definition 2.2.4. Let M, N be differential manifolds and consider a differ- ential map f : M → N . then one can define a differential map

df : T

→ T

as follows. Take a tangent vector τ ∈ T

, and let P ∈ M be the application point of τ . Fix one chart (U, φ) of M which contains P , and fix one chart (V, ψ) of N which contains f (P ). Let v be the vector part of τ in (U, φ) and consider the map

α = ψ ◦ f ◦ φ

: φ(U ∩ f

(V )) → ψ(V ).

Then define df (τ ) as the tangent vector applied in f (P ), whose vector part in (V, ψ) is w = d(α)

(v).

The definition is not complete, unless we prove that df (τ ) is independent from the choice of the charts (U, φ), (V, ψ). To do that, fix another chart (U

, φ

) of M containing P . The vector part of τ in the new chart is v

= d(φ

◦ φ

(v). Since differentials preserve compositions, we get:

d(ψ ◦ f ◦ φ

)

(v

) = d(ψ ◦ f ◦ φ

)

(d(φ

◦ φ

(v)) =

= d(ψ ◦ f ◦ φ

◦ φ

◦ φ

)

(v) = d(ψ ◦ f ◦ φ

)

(v) = w.

This proves that the choice of the chart in M is not relevant in the definition of df (τ ). The proof that the definition does not depend on the choice of the chart in N is similar, and it is left to the reader as an exercise.

Some elementary properties of the differential df are listed below.

Proposition 2.2.5.

a) df is a differential map.

b) If f : M → M

and g : M

→ M

are differential maps, then d(g ◦ f ) =

dg ◦ df .

c) The differential of the identity on M is the identity on T

.

d) If f is a diffeomorphism, then also df is a diffeomorphism, and d(f

) = (df )

.

Proof. b), c), and d) are an easy consequence of the corresponding properties of the differential of a C

map between two open subsets of R

, R

. We leave the proof as an exercise.

To prove a), fix charts (U, φ), (V, ψ) of M, N respectively, and take the corresponding charts (π

(U ), Φ), (π

(V ), Ψ) of T

, T

respectively. We need to prove that

Ψ ◦ df ◦ Φ

: φ(U ∩ f

(V )) × R

→ R

(where m = dim(M ) and n = dim(N )) is of class C

. Notice that if (Q, v) ∈ φ(U ∩ f

(V )) × R

⊂ R

, then Φ

(Q, v) is the tangent vector applied in Q, whose vector part in (U, φ) is v. Thus

Ψ ◦ df ◦ Φ

(Q, v) = (ψ ◦ f ◦ φ

(Q), d(ψ ◦ f ◦ φ

)

(v)).

The conclusion follows from Lemma 2.1.7.

By definition, if τ is applied in P , then df (τ ) is applied in f (P ). Thus the following diagram is commutative.

T

−−−−−→

T

π ↓ ↓ π

M −−−−−→

N

Remark 2.2.6. If we fix P ∈ M , then the differential df determines a

linear map between the vector spaces T

and T

. Namely, for every

choice of the charts (U, φ) and (V, ψ) containing P and f (P ) respectively, the

differential of ψ ◦ f ◦ φ

, computed in φ(P ), acts linearly on the vector parts

of vectors in T

. We will denote with df

the restriction of df to T

.

Definition 2.2.7. We say that a differential map f : M → N is an immer- sion if f is injective and also the differential df : T

→ T

is injective.

We say that M is a submanifold of N if there exists an immersion f : M → N .

Example 2.2.8. If M is a differential manifold with only one chart, then by definition it is immediate to realize that T

is diffeomorphic to M × R

(m = dim(M )).

In particular, if M = R

and N = R

, the the notion of differential of a map f : M → N corresponds, on the vector parts, to the usual differential of C

function, defined in the courde of Calculus.

We notice immediately, as we will see in examples, that when M has more than one chart, then T

and M × R

are not necessarily diffeomorphic.

Example 2.2.9. Consider the function f : R → R

, f (t) = (t

, t

). Even if f is C

and injective, yet it is not an immersion. Indeed, the jacobian matrix of f in the point t = 0 vanishes, so df sends every vector in T

to the null vector.

Next examples show that the tangent spaces to varieties with an immersion

to R

behave as the tangent spaces of euclidean geometry.

Example 2.2.10. Let C be the circle, with the differential structure defined in Example 1.1.7 and let f : C → R

be the injective map which sends every point of the circle to the point itself, considered as a point in R

. With the notation of Example 1.1.7 consider the composition α = (id

◦ f ◦ φ

), defined in the image of the chart (U, φ). One verifies that

α(z) =

 4z

4 + z

, 4 − z

4 + z

 , so that one computes

(dα)

=  16 − 4z

(4 + z

)

, −16z (4 + z

)

 .

It is clear that for no values of z ∈ R \ {0} the matrix of (dα)

vanishes, thus for all z in the chart the map(dα)

is injective. A similar computation holds in the second chart of C. Thus f is an immersion.

Let Q be a point of the chart (U, φ) and let τ be a tangent vector applied in Q, whose vector part in (U.φ) is t ∈ R. If Q corresponds to the point of coordinates (a, b) in the plane, then one computes, with easy arguments of analytic geometry, that φ(Q) = 2a/(1 − b). Thus, setting z = 2a/(1 − b) in the previous formulas, one gets

(dα)

(t) =  b(b − 1)

2 t, a(1 − b)

2 t

 .

Thus (df )

(τ ) is a vector of R

applied in Q, whose vector part is orthogonal to the radius passing through Q. It follows that the image of T

in df corresponds to the tangent line to the circle in Q, of the euclidean geometry.

Example 2.2.11. Let S be a surface in R

implicitly defined by the vanishing

of a C

function F (x, y, z) = 0. Assume that the gradient (

,

,

) is

nowhere zero in S. Then we give to S the differential structure of Example

1.1.8. The inclusion g : S → R

becomes an immersion of S into the euclidean

space.

Namely, for any P = (x

, y

, z

) ∈ § there exists a chart (U, φ) of S, containing P , such that U corresponds to the graph of a C

function f : Ω → R

, where Ω ⊂ R

is a open neighborhood of φ(P ). If, for instance, ∂F/∂z is nowhere zero in U , then we may assume that φ is the projection to the first two coordinates, and z = f (x, y) for all points of U .

Consider a tangent vector τ , applied at P , whose vector part in (U, φ) is v ∈ R

. Then the vector part of (dg)(τ ) in the unique chart of R

is (dα)

(v), where α is the map α = (id

) ◦ g ◦ φ

. Hence α(x, y) = (x, y, f (x, y)), so that (dα)

(v) is the product of the matrix





1 0

0 1

∂f /∂x ∂f /∂y





(x0,y0)

times v. Since the matrix has always rank 2, then dg is injective in U . The same holds in any chart of S, so that g is an immersion.

Notice that, when τ ranges in T

, dg(τ ) describes the plane orthogonal to the vector (∂f /∂x)

, (∂f /∂y)

, 1). By the Implicit Function Theorem, one knows that

(∂f /∂x)

= (∂F/∂x)

/(∂F/∂z)

(∂f /∂x)

= (∂F/∂y)

/(∂F/∂z)

,

hence the image of T

corresponds to the plane passing through g(P ) and orthogonal to the gradient of F in P , which is the usual tangent plane to S in the euclidean geometry.

The conclusions of the last two examples hold in general: every time that a differentil manifold M has an immersion in R

, the image of the tangent spaces to M correspond to affine linear subspaces that coincide with the intuitive euclidean definition of tangent spaces to geometric objects in R

.

The Inverse Function Theorem has an analogue for differential maps.

Theorem 2.2.12. Let f : M → M

be a differential map. If for some P ∈ M

the linear map (df )

: T

→ T

is invertible, then f is invertible in a