SJ ¨ ALVST ¨ ANDIGA ARBETEN I MATEMATIK
MATEMATISKA INSTITUTIONEN, STOCKHOLMS UNIVERSITET
Introduction to Differential Manifolds and their Cohomology
av
Jorina Marlena Sch¨utt
2012 - No 20
Introduction to Differential Manifolds and their Cohomology
Jorina Marlena Sch¨utt
Sj¨alvst¨andigt arbete i matematik 30 h¨ogskolepo¨ang, avancerad niv˚a Handledare: Boris Shapiro
Abstract
This exposition starts with basic algebraic definitions such as module, tensor product, tensor algebra, as well as symmetric and skew-symmetric algebras. Next, we define n-dimensional manifolds as topological spaces locally homeomorphic to balls inRn. In the smooth case we define di↵erential forms of arbitrary orders. Then we will prove the fundamental Stokes theorem for di↵erential forms, which, in particular, explain how a surface integral of a vector field over an oriented surface is related to the volume integral of its divergence over the body bounded by the surface. We will investigate Stokes theorem for cuboids, simplices and general manifolds. Finally, we define the notion of de Rham cohomology of a smooth manifold using the famous Poincar´e lemma. De Rham cohomology is a analytical way of approaching the algebraic topology of a manifold. De Rham theorem claims that de Rham cohomology group of a manifold is isomorphic with its singular homology group.
For purposes of illustration, we provide a connection between the vector analytic notions such as gradient, divergence and curl in R3 and the singular homology of the corresponding objects. At the end we will take a look at Morse inequalities. Morse theory gives a direct way of analyzing the topology of a manifold by studying smooth real-valued functions on it.
Sammanfattning
Detta examensarbete utg˚ar ifr˚an grundl¨aggande definitioner s˚asom modul, tensorprodukt, tensoralgebra, b˚ade symmetriska och
asymmetriska algebra. D¨arefter definierar vi n-dimensionella m˚angfalder som topologiska rum lokal homeomorfa till bollar iRn. Sedan bevisar vi den fundamentala satsen, Stokes sats, f¨or
di↵erentialformer, som framf¨or allt f¨orklarar hur en ytintegral av ett vektorf¨alt ¨over en orienterad yta h¨anger ihop med volymintegralen
¨
over dess divergens ¨over kroppen avgr¨ansas av ytan. Vi betraktar satsen f¨or b˚ade block, simplex och generella m˚angfalder. Vi definierar begreppet de Rham kohomologin under anv¨andandet av den v¨alk¨anda Poincar´e lemma. De Rham kohomologin ¨ar ett analytiskt s¨att att n¨arma sig algebraiska topologin av en m˚angfald.
De Rham teorem p˚ast˚ar att de Rham kohomologi gruppen av en m˚angfald ¨ar isomorfisk med dess singul¨ar homologi grupp. Vi etablerar, f¨or illustrations¨andam˚al, sambandet mellan vektor analytiska begrepp s˚asom gradient, divergens och rotation iR3 och den singul¨ara homologin med motsvarande objekter. Mot slutet av arbetet betraktar vi Morse olikheter. Morse teorin ger ett direkt s¨att f¨or att analysera topologin av m˚angfalder genom att studera di↵erentialfunktioner p˚a dem.
Contents
1 Elements of Linear Algebra
with focus on Tensor products 5
1.1 Modules . . . 5
1.1.1 Left and Right modules . . . 5
1.1.2 Examples of R-modules . . . 6
1.1.3 R-module homomorphisms . . . 6
1.2 Tensor product . . . 6
1.2.1 Universality of tensor product . . . 7
1.3 Tensor algebra . . . 10
1.3.1 Symmetric algebra . . . 11
1.3.2 Skew-symmetric algebra . . . 12
2 Di↵erentiable Manifolds 13 2.1 Topological n- dimensional manifolds . . . 13
2.2 Local coordinates on topological manifolds . . . 14
2.2.1 Di↵erentiable Manifolds . . . 15
2.3 Atlas . . . 16
3 Tangent and Cotangent vectors and Di↵erential forms 19 3.1 Smooth maps . . . 19
3.2 Tangent vector . . . 21
3.3 Tangent bundle . . . 22
3.4 Vector field . . . 22
3.5 Cotangent space to M at p . . . 24
3.6 Di↵erential forms . . . 25
3.6.1 1-forms . . . 25
3.6.2 Di↵erential q-forms on M . . . 27
3.6.3 Di↵erential n 1-forms on M . . . 29
3.7 ⇤ - operation . . . 31
3.7.1 Pullback of Di↵erential forms . . . 31
4 Integration of Di↵erential forms and Stokes Theorem 34 4.1 Orientation of submanifolds . . . 34
4.2 Integration of a q-form . . . 37
4.3 Stokes theorem . . . 40
4.3.1 Stokes Theorem . . . 40
4.3.2 Using holomorphic functions . . . 42
5 De Rham Cohomology and its relation to vector cal- culus 46 5.1 De Rham cohomology complex and cohomology groups . . . 50
5.2 Properties of de Rham cohomology . . . 52
5.3 Simplicial homology . . . 59
5.4 Main result - de Rham Theorem . . . 67
6 Introduction to Morse theory 69 6.1 Basic concept . . . 69
6.2 Formal preparation . . . 71
6.3 Basics of Morse theory . . . 73
6.4 Morse inequalities . . . 74
1 Elements of Linear Algebra with focus on Tensor products
1.1 Modules
The notion of a vector space can easily be generalized by permitting instead of a field a ring R as the set of scalars. Doing so we introduce a so called R-module. For example, every abelian group is aZ-module.
Besides modules we will define in this section the tensor product of modules and finally the tensor algebra.
I refer mainly to [KER], [STU] and [BOS] in this section.
1.1.1 Left and Right modules
Let R be a commutative ring.
Definition 1. 1. We call an operation R⇥ M ! M, (r, m) 7! rm where R acts on an abelian group M by scalar multiplication a R-module or left R-module (notation R mod) if for all r1, r2, r2 R and m1, m2, m2 M the following holds:
r· (m1+ m2) = rm1+ rm2
(r1+ r2)m = r1m + r2m (⇤) r1· (r2m) = (r1r2)· m
1· m = m
2. We call an operation M ⇥ R ! M, (m, r) 7! mr where the abelian group M acts on R by scalar multiplication a right R- module (notation mod R) if properties analogous to (⇤) hold.
1.1.2 Examples of R-modules
1. R itself is an R-module (with multiplication in R);
2. Every K-vector space, where R = K is a field;
3. Every abelian group G is a Z-module: all n terms of nx = x + ... + x lie in G and therefore ( n)x = (nx) lies in G for every x2 G and n 2 N.
1.1.3 R-module homomorphisms
Definition 2. A mapping ' : M ! M0 with an R-module M, M0 is called R-module homomorphism or R-linear, if for m, m02 M, r 2 R the following holds:
'(m + m0) = '(m) + '(m0), '(rm) = r'(m).
In the same way we can generalize the notion of a K-algebra and obtain a R-algebra. For example, the set of all endomorphisms of a R-module EndRM := {' : M ! M|' is R-linear} is an R-algebra with the properties:
(' + )(m) := '(m) + (m), (' )(m) := '( (m)),
(r')(m) := r'(m), for all m2 M, r 2 R, ', 2 EndRM .
Definition 3. An additive subgroup N of an R-module is called its submodule, if rn2 N for all n 2 N, r 2 R.
1.2 Tensor product
Let R be a ring, M be a right-R-module, and N be a left-R-module.
Let U be the submodule ofZM⇥N; i.e. U is generated by the elements of the following form:
(m + m0, n) (m, n) (m0, n), (m, n + n0) (m, n) (m, n0),
(mr, n) (m, rn), where r2 R, m, m02 M and n, n02 N.
Definition 4. Define the tensor product of M and N over R by ZM⇥N/U =: M⌦RN . In other words, for some elements m2 M and n2 N we define m ⌦ n as the residue class (m, n) + U in M ⌦RN .
For all r2 R, m, m02 M, n, n02 N the following properties hold:
•
(m + m0)⌦ n = m ⌦ n + m0⌦ n, m⌦ (n + n0) = m⌦ n + m ⌦ n0,
mr⌦ n = m ⌦ rn;
• every element z 2 M ⌦RN has a representation of the form
z = Xj i=1
mi⌦ ni where mi2 M, ni2 N, j 2 N.
This decomposition is, in general, non-unique.
• If R is commutative, then M ⌦RN has the property:
r(m⌦ n) = mr ⌦ n = m ⌦ rn.
Example 1. For M =R2and N =R3the tensor product T = M⌦ N is given by 2⇥ 3 real valued matrices. Take x = (x1, x2) and y = (y1, y2, y3) inR2andR3respectively. The tensor product combines x and y in the rank 1 matrix xyT:
x⌦ y = xyT
✓x1
x2
◆
⌦ y1 y2 y3 =
=
✓x1y1 x1y2 x1y3
x2y1 x2y2 x2y3
◆
= X2 i=1
X3 j=1
xiyjEi,j,
where Ei,j is the 2⇥ 3 matrix with 1 at the (i, j)th entry and 0 else.
The combinations of rank 1 matrices give all 2 times 3 matrices and thus the dimension of T =R2⌦ R3is 6.
1.2.1 Universality of tensor product
Theorem 1. Let V be a Z module. Then every bilinear mapping : M⇥ N ! V induces a unique homomorphism
g : M⌦RN! V satisfying g(m ⌦ n) = (m, n).
In other words, the following diagram is commutative:
M⇥ N
⌦
✏✏ //V
M⌦RN
9!g
u::u uu uu uu uu
Proof. By bilinearity of we have
(mr, n) = (m, rn) for all m2 M, n 2 N, r 2 R. (1) Define
g : (M⌦ N = ZM⇥N/U )! V ((m⌦ n) = U + (m, n)) 7! (m, n),
where U is the submodule ofZM⇥N with U ⇢ ker(g) defined as above.
• well-definedness: indeed, consider U + (m, n) = U + (m0, n0) for (m, n), (m0, n0) 2 M ⇥ N, then there exists u 2 U with (m, n) = u + (m0, n0). It follows g((m, n)) = (u + (m0, n0)) = (u) + ((m0, n0)). Since u 2 U ⇢ ker(g) we get g((m, n)) = g((m0, n0)).
• g is a homomorphism:
(i) for (m, n), (m0, n0)2 M ⇥ N we have:
g((U + (m, n)) + (U + (m0, n0)))
= g(U + ((m, n) + (m0, n0))) =
= ((m, n) + (m0, n0)) =
= ((m, n)) + ((m0, n0)) =
= g(U + (m, n)) + g(U + (m0, n0));
(ii) for (m, n)2 M ⇥ N, r 2 R we have:
g((U + (m, n))r) = g(U + (m, n)r) =
= ((m, n)r) = (m, n)r =
= g(U + (m, n))r.
• commutativity: g ⌦ = holds because by construction we have for all (m, n)2 M ⇥ N:
g ⌦(m, n) = g(m ⌦ n) = g(U + (m, n)) = (m, n).
• uniqueness: let h be a homomorphism with h ⌦ = , then for each (m, n)2 M ⇥ N we have:
h(U + (m, n)) = h ⌦(m, n) =
= (m, n) = g ⌦(m, n) =
= g(U + (m, n)) ) h = g.
Theorem 2. Tensor product induces two canonical Z-isomorphisms:
M ⌦RR! M, m ⌦ r 7! mr, R⌦RN ! N, r ⌦ n 7! rn.
Proof. Take the bilinear mapping : M ⇥ R ! M, (m, r) 7! mr, which satisfies equation (1). By Theorem 1 (universality of tensor product) there exists exactly oneZ-linear mapping g : M ⌦RR! M with g(m⌦ r) = mr for all m 2 M, r 2 R. This mapping is bijective and its invertion is given by M ! M ⌦RR, m7! m ⌦ 1.
Lemma 1. (Tensor product of vector spaces)
Let K be a field and M = V and N = W be vector spaces over K. Then V ⌦K W is a K-vector space with elements of the form v⌦ w =P
i2I,j2J iµj(ei⌦ fj) where E ={ei|i 2 I} a basis to V (i.e.
v =P
i2I iei), F ={fj|j 2 J} a basis to W (i.e. w =P
j2Jµjfj).
Furthermore, dim(V ⌦ W ) = dim V · dim W .
Proof. We can interprete tensor product of finite-dimensional vector spaces as the space of matrices, where we determine the rows with index I ={1, ..., n} and the columns with index J = {1, ..., m}. Then the entries of the columns are the multiplicities of v =P
i2I ieiand the rows multiplicities of w =P
j2Jµjfj. Tensor multiplication v⌦ w has the following properties:
(v0+ v00)⌦ w = v0⌦ w + v00⌦ w, v⌦ (w0+ w00) = v⌦ w0+ v⌦ w00,
( v)⌦ w = (v ⌦ w) = v ⌦ ( w), where v, v0, v002 V, w, w0, w002 W, 2 K.
In general, commutativity does not hold, because for v2 V, w 2 W the vectors v⌦ w 2 V ⌦ W and w ⌦ v 2 W ⌦ V lie in the same space only in case V ⌘ W . Even then the equality v ⌦ w = w ⌦ v is not necessarily true. Constructing unique ordered pairs
E⇥ F = {(ei, fj)|i 2 I, j 2 J} from the initial two bases
E ={ei|i 2 I} and F = {fj|j 2 J} as cartesian products we get that the dimension of V ⌦ W equals the product of the dimensions of V and W .
1.3 Tensor algebra
Definition 5. Let R be a commutative ring with an unit element and let M, N be R-modules. By definition M⌦RN is a R-module.
We define the i-fold tensor product by M⌦i= M⌦RM⌦R...⌦RM
| {z }
itimes
. Note: for i = 0 we get M⌦0= R and for i = 1M⌦1= M .
Now we are able to define the tensor algebra T (M ) =L
i 0M⌦i= R M (M⌦ M) (M⌦ M ⌦ M) ..., where denotes the direct sum.
Example 2. Considering for R = R and M = R2 we get: T (R2) = Ln
i=0Ti(R2) =Ln
i=0(R2)⌦i=R R2 (R2⌦ R2) .... Expansion of the first three summands we can obtain bases: R = hxi, R2=hx, yi, R2⌦ R2=hxx, xy, yx, yyi, since
✓x y
◆
⌦
✓x y
◆
=
✓xx xy yx yy
◆ .
Lemma 2. 1. Let R be a commutative ring and M1, M2 be R- modules. Then
T (M1 M2)⇣ T(M1)⌦ T (M2) defines an isomorphism, which we call canonical.
2. Let V be a vector space and let V⇤ = Hom(V, K), the set of homomorphisms from V to K, be its dual space. Then there is an isomorphism:
T (V )⌦ T (V⇤) ⇠= (M
r 0
V⌦r)⌦ (M
s 0
(V⇤)⌦s)
= M
r,s 0
(V⌦r⌦ (V⇤)⌦s)
In other words, an arbitrary element x in T (V )⌦ T (V⇤) is rep- resented by
x = X
r,s 0
xr,s ; where xr,s2 V⌦r⌦ (V⇤)⌦s.
Note: x1,0 2 V ⌦ K is a vector and x0,12 K ⌦ V⇤ is a linear form.
Proof. 1. Follows similiarly as in theorem 2 in section 1.2.1.
2. Consider the commutative diagram:
V
f
✏✏
a //A
T (V )
9! ˜f
<<z zz zz zz z
f˜ a = f : ˜f (v1⌦ v2⌦ ..) = f(v1)· f(v2)· ...
where f is a linear map, a the direct sum of the i-fold tensor product and A is aK-algebra.
1.3.1 Symmetric algebra
Definition 6. Let M⌦n be a n-fold tensor product of a R-module M and let T (M ) be the corresponding tensor algebra. Consider the ideal X in T (M ) generated by
[m1, m2] = m1⌦ m2 m2⌦ m1,
where m1, m22 M. We define the symmetric algebra of M as S(M ) = T (M )/X =M
n 0
Sn(M ) = R M S2(M ) ...,
where Sn(M ) = Tn(M )/X.
Example 3. Considering for R = R and M = R2 we get: S(R2) = Ln
i=0Si(R2) =Ln
i=0(R2)⌦i=R R2 (R2⌦ R2) .... Expansion of the first three summands we can obtain bases: R = hxi, R2=hx, yi, R2⌦ R2=hxx, xy, yyi, since
✓x y
◆
⌦
✓x y
◆
=
✓xx xy yx yy
◆
and xy = yx.
1.3.2 Skew-symmetric algebra
Definition 7. Let again M⌦n be a n-fold tensor product of a R- module M and let T (M ) be the corresponding tensor algebra. Con- sider the ideal Y in T (M ) generated by
[m, m] = m⌦ m,
where m2 M. We define the skew-symmmetric (or Grassmann alge- bra) of M as
⇤(M ) = T (M )/Y =M
n 0
⇤n(M ) = R M ⇤2(M ) ...,
where ⇤n(M ) = Tn(M )/Y is the so called n-th exterior power of M .
Definition 8. In the above notations the exterior product^ of two elements m1, m22 ⇤(M) is defined as
m1^ m2= m1⌦ m2(modY ).
Properties of the exterior power, mi2 M for i = 1, ..., n:
1. m1^ m2^ ... ^ mn= (m1⌦ m2⌦ ... ⌦ mn) =
= (m1⌦ m2⌦ ... ⌦ mk)⌦ (mk+1⌦ mk+2⌦ ... ⌦ mn), 2. mi= mj ) mi^ mj= 0,
3. mi^ mj= mj^ mi.
Example 4. Skew-symmetric algebra on vector spaces.
In particular, we can define the skew-symmetric algebra of a vector space V over a field K. Consider the index set I = {i1, i2, ..., in|i1 ... in} and the generating system, eI= ei1^ ei2^ ... ^ ein. Then:
⇤(V ) = M
I2{1,...,n}
KeI.
For two vectors v = (v1, v2)T = v1e1+ v2e2 and w = (w1, w2)T = w1e1+ w2e2 in ⇤(V ) the exterior product is given as follows:
v^ w = (v1e1+ v2e2)^ (w1e1+ w2e2)
= v1w1e1^ e1+ v1w2e1^ e2+ + v2w1e2^ e1+ v2w2e2^ e2
= (v1w2 v2w1)e1^ e2.
2 Di↵erentiable Manifolds
Di↵erentiable manifolds are higher dimensional analogues of surfaces.
We should, nevertheless, not think of a manifold as an object, which always sits inside an Euclidian space, but rather as an abstract object.
After the definition of a di↵erentiable manifold we want to describe points locally by n real numbers, local coordinates.
I refer mainly to [STU], [WAR] and [HOF] in this section.
2.1 Topological n- dimensional manifolds
We begin with some notions of topology. For our considerations it is enough to refer to [HIS].
Definition 9. Let X be a set and P (X) its power set. A topology T is a family of sets, open subset of P (X), with the following properties:
• ;, X are open sets,
• the intersection of finitely many open sets is open,
• any union of open sets is open.
Then one calls the pair (X, T ) a topological space.
Definition 10. A system B of subsets of (X, T ) is called basis of topology, if
• every open set of B is open with respect to T ,
• every open set of (X, T ) is representable as a union of sets of B.
Here one understands the empty union as the empty set;.
Definition 11. A topological space (X, T ) satisfies the second axiom of countability, if and only if X has a countable basis consisting of open sets.
Now we can define the term of topological n-dimensional manifold.
Definition 12. We call a set M a topological n-dimensional manifold, if M is a topological space such that for each point x2 M there exists an open neighborhood U (x) and a locally bijective and continuous map ':
U '! V ⇢ Rn
Remark 1. A connected 1-dimensional topological manifold is home- omorphic to a segment of a straight line.
Picture 1: topological n-dimensional manifold
Remark 2. Submanifolds are subsets of manifolds, which are self man- ifolds. Generally speaking, a submanifold of a manifold is what a subspace is of a vector space; only that submanifolds and manifolds are of the same dimension. Suppose, for example, the Earth’s surface as a manifold, then a meridian is a submanifold.
2.2 Local coordinates on topological manifolds Let M be a manifold.
Definition 13. Given an open covering M =S
i2IUi and open sets Vi 2 Rn, we call the homeomorphisms 'i : Ui ! Vi local charts on M . The 'i determine the change of local coordinates on Ui to local coordinates on Vi.
To be able to describe properties of a manifold M , which is as mentioned earlier, somehow a object without reference system, we want to transfer M to coordinates we can work better with, namely coordintes inRn. Not to distort everything this change of local coordinates has to happen smooth with help of local charts.
Definition 14. We call the following smooth composition of two maps change of local coordinates:
'i 'j1: Vji(= 'j(Ui\ Uj))! Vij(= 'i(Ui\ Uj)) where both maps act on the subset Ui\ Uj⇢ M.
Picture 2: change of local coordinates
2.2.1 Di↵erentiable Manifolds
Definition 15. A topological manifold is said to be Ck-di↵erentiable, respectively C1-di↵erentiable or smooth, if and only if
(i) M satisfies the 2nd axiom of countability and
(ii) the local change of coordinates on M is Ck, C1, respectively.
2.3 Atlas
As for world atlases, we need several charts to depict the world and the more charts we take, the better is the representation of reality.
Nevertheless, we need to assure that the charts are compatible with each other; that we can smoothly stick the charts together to one big atlas.
Definition 16. An atlas is the set of Ui ⇢ M with the local charts 'i: Ui! Viand i2 I an index set: A = {(Ui; 'i)|i 2 I}.
Definition 17. Let U, Ui, Ui0⇢ M and V, Vi, Vi0⇢ Rnbe open subsets.
• A map ' : U ! V is called compatible with a given atlas A on a smooth manifold M if and only if the operation ' 'i1 : 'i(U\ Ui)! '(U \ Ui) is a di↵eomorphism (i.e. ' 'i1 and its inverse are smooth).
• Two atlases A = (Ui; 'i: Ui! Vi), A0= (Ui0; '0i: Ui0! Vi0) are called compatible on M if and only if
(i) every chart of (Ui, 'i) is compatible with A0 and (ii) every chart of (Ui0, '0i) is compatible with A.
Definition 18. We call the union of all maps which are compatible with a given atlas A the maximal Atlas A =S
'i2AA.
Remark 3. It follows from the defintion that every atlas A is contained in the unique maximal atlasA.
Remark 4. (Refinement of open covers)
Take an open coverU = (Ui|i 2 I) of a manifold M. Let the open sets V = (Vj|j 2 I) be a refinement of U. This means, V is a new cover of M such that every set inV is contained in some set in U. We can select the indices by the function ↵ : J ! I ; j 7! i: then Vj⇢ U↵(j). This is motivated by the Heine-Borel Theorem claiming that: if the set M is covered by the union of open coversS
i2IUi◆ M then there exists a subindex set J ⇢ I, so is M even representable by the union over the new indexed covers S
j2JUj ◆ M. In other words, every open cover of a compact subset of Rn has a finite subcover. See for further considerations [RUD].
Theorem 3. (Heine - Borel)
For a subset S ofRn the following statements are equivalent:
(a) S is closed and bounded;
(b) every open cover of S has a finite subcover (i.e. S is compact).
Proof. (a) ) (b) (closed and bounded implies compact): Assume that S is bounded. We can cover S ⇢ Rn by the Cartesian product of n intervals [ai, bi], ai < bi, ai, bi 2 R , i = 1, ..., n: S ✓ T0 = [a1, b1]⇥[as, b2]⇥...⇥[an, bn]. We can divide each side of T0into halves.
Doing so we get 2n smaller cells. To get a contradiction we assume that T0 is not compact, i.e. T0 is covered by open sets {G↵} which have no finite subcover. Now take the 2nsmaller cells into account: at least one of those cannot be covered by a open subcellection of{G↵}.
Otherwise the whole T0would be covered in this way. Call this small cell T1. Considering T1 we divide its sides in halves again and pick out the next cell T2, which is not covered by a open subcollection of {G↵}. Continuing this process we construct a sequence of cells with the following properties:
(i) T0 T1 T2 ... Tk ...;
(ii) Tk is not covered by any finite subcollection of{G↵};
(iii) the length of all sides Tk tends to zero when k goes to1, i.e.:
limk!1bk2kak = 0 By Cantor’s lemma we getT1
k=1Tk6= ;, which means that there is a point p in the intersection of all enclosed cells, p2 Tk for all k 2 N.
Because G↵ is open there exists an open ball B(p) around p. Since p sits in all Tk this ball B(p) works as the finite subcover. Since we covered S by the n-cell T0we can make a selection of open balls cov- ering S and we get a contradiction.
(b)) (a) (compact implies closed and bounded): (1) Compact implies closed: Taken a point y in the complement to S: y2 SC. For all x2 S there exist nonintersecting neighborhoods Bx
containing x andByxcontaining y. The union of allBxbuilds an open cover of S, S⇢S
x2SBx. Since S is assumed to be compact, there exists a open, finite subcoverBx1, ...,Bxn of S. Consider the neighborhood of y, which lies outside S: Tn
k=1Bxyk. This means that y (any point outside S) can´t be a limit point of S and therefore all limit points must already lie inside S.
(2) Compact implies bounded: Consider an open ball centered at a common point of S of any desired radius. Because all points have the
same distance to the ball´s boundary, it can cover any set. Since all balls in the subcover are contained in the largest open ball entirely lying in that subcover, they must be bounded. Therefore the set S which is covered now by the bounded smaller balls, must be bounded as well.
3 Tangent and Cotangent vectors and Di↵er- ential forms
In this section M is a di↵erentiable manifold and let A ={(Ui; 'i)|i 2 I} be a smooth atlas on M with local charts 'i : Ui ! Vi, where Ui✓ M and Vi✓ Rn.
I refer mainly to [FRI],[FOR],[GSI] and [HIT] in this section.
3.1 Smooth maps
• Let U ⇢ M be an open subset and let p be a point in U. We say that the map f : U ! Rn is smooth at p if and only if for any chart 'i the composition f 'i1is smooth at 'i(p), i.e.
f : U! Rn is smooth at p2 U
, for each chart 'ithe composition f 'i1 is '(Ui\ U) ! Rn is smooth in 'i(p).
Ui\ U
f
✏✏
'i
//Vi f 'i1
{{wwwwwwwww Rn
• Let M1 and M2 be manifolds, Ui ⇢ Mi and Vi ⇢ Rn be open subsets for i = 1, 2. We say that the map f : M1 ! M2 is smooth at p if and only if the composition of f and two charts 'i: Ui! Vi, f (U1)⇢ U2, '2 f|U1 '11: V1! V2is smooth; i.e.
f : M1! M2is smooth
, for each two charts 'i : Ui ! Vi, Ui ⇢ Mi, i = 1, 2, f (U1)⇢ U2, the restriction '2 f|U1 '11: V1! V2 is smooth.
M1 U1 f|U1
//U2⇢ M2 '2
✏✏V1 '11
OO
'2_f_|U1_'_11_//
_ V2
Definition 19. We call f : M1! M2 a di↵eomorphism, if (i) f is a homeomorphism: f is bijective, f, f 1 are continuous, (ii) f, f 1 are C1-di↵erentiable.
Even if we define manifolds with the help of local charts, there is no chart marked as ’more preferably’ than another. According to this, the choice of charts chould later be rather unimportant. We define a smooth function, which maps every point p on a manifold M to a real number. Then we get, collecting all these functions, a set of all smooth functions from M toR.
Definition 20. We define the ring of smooth functions by C1(U ) ={f : M ! R|f is smooth for all p 2 M}.
This definition is given without local charts and this is meaningsfull, because it is possible to define a map from any set to the real numbers. This means, in other words, that the value of a function is independent of the choice of local coordinates.
Remark 5. • For any p 2 U consider the set Mp={f 2 C1(U )|f(p) = 0}. This is a maximal ideal in C1(U ).
• C1(U ) is aR-vector space.
Proof. • By definition a maximal ideal of a ring R is an ideal M satisfying:
for each ideal a2 R with M ⇢ a ⇢ R : either a = M or a = R Denote by ev0 the mapping which evaluates every function at the origin. The image under ev0 is R and its kernel, the set of all f (p) = 0, is a maximal ideal.
• The axioms of a vector space are easily verified.
These functions have many applications, for example, the electrical potential can be understood as such a scalar field. But since the electrical and the magnetic fields are attracting each other, we need to introduce a reference system, which respects both forces and specifies both, a scalar and a direction at a point p on a manifold M . We need later on a vector field.
3.2 Tangent vector
Because we are not longer in a Euclidean space but in a curved space, we can´t define vectors globally. Each point p of a manifold M has its own vector space.
Definition 21. Let M be a di↵erentiable manifold and p2 M be a point on M . Define the vector X on M at p as the map from CM,p1
to Rn, mapping every smooth function f 2 CM,p1 on a di↵erentiable manifold M to a real number, which satisfies the following properties:
• X is a R-linear mapping: ( X1+ ⌫X2)f = (X1f ) + ⌫(X2f )
• X respects the Leibnitz rule:
X(f1+ f2) = X(f1)f2(p) + f1(p)X(f2)
We can consider vectors as directional derivatives. They specify the rate of change of a scalar field in the direction they point.
Definition 22. For a function f : M ! Rn and the on M in local coordinates represented points p = (p1, p2, ..., pn), h = (h1, h2, ..., hn) the directional derivative is defined as:
Dh(f ) = lim
t!0
f (p + th) f (p)
t =
Xn i=1
hi @
@xi|p(f ).
In the special case: hi = ei the unit vectors: Dei(f ) = @x@
i|p(f );
i = 1, ..., n, we get the usual partial derivative.
The direction a vector points can be determined by calculating the derivative in direction of a curve: let k :R ! M be a smooth curve and let p2 Mbe a point on M with p = k(tp) for tp2 R. The map which at a given point p assigns to each function f 2 CM,p1 the real number
df (k(t)) dt |t=tp
is called tangent vector. We note that this coincides due to the above defintion 21 of a vector with the general derivative and we define finally
X :CM,p1 ! R f 7 !df (k(t))
dt |t=tp
as the tangent vector at p = k(tp) of a smooth curve k :R ! M, tp2 R.
Given a tangent vector X at p, we define the set of tangent vectors to M at p as TM,p={X|X tangent vector in p}.
Note:
• TRn,p=R@x@1|p+R@x@2|p+ ... +R@x@n|p
• TSn 1,p={Pn i=1 i @
@xi|Pn
i=1 ipi= 0} ✓ TRn,p
Here we denote the (n 1)-dimensional unit sphere:
Sn 1={x 2 Rn| kxk = 1}
For f (x) = x21+ ... + x2n 1 is
Sn 1={x 2 Rn|f(x) = 0}
and further is gradf (x) = (2x1, ..., 2xn) implying that gradf (x) 6= 0 for all x2 Sn 1.
3.3 Tangent bundle
Definition 23. Given a smooth manifold M we define its tangent bundle T (M ) as the union of tangential vectors over all points p on the manifold M :
T (M ) = [
p2M
TM,p
Although we need later charts with local coordinates on manifolds M to define di↵erentiablility of functions, which are acting on M , we have defined the term of vector independent of coordinates. Further, our notion of a vector got the vivid idea of a direction, which might be defined by a curve.
3.4 Vector field
Until now we have no possibility to compare elements of the tangent space at one point with the tangent space at another point. Before we get rid of this problem we first define a vector field.
Definition 24. We define a vector field for each point p on a smooth manifold M as the mapping from M to the tangent bundle T (M ). A point p2 M is mapped to its vector V (p) 2 TM,p:
V : U ! T (M) p7 !
Xn i=1
i(p) @
@xi|p
Note: We call V smooth if and only if the generating functions i
are smooth for i = 1, ..., n
Theorem 4. Consider a k-dimensional manifold M and a point p2 M , then:
(a) TM,p is a k-dimensional subspace inRn.
(b) Consider a map ' : M ! V , where V ⇢ Rk open and a point p2 U. The vectors @t@'1(p), ...,@t@'
k(p) form a basis for TM,p. (c) Let N ⇢ Rnbe an open neighborhood of p and f1, ..., fn k: N !
R smooth functions with
M\ N = {x 2 N : f1(x) = ... = fn k(x) = 0} and rank@(f1, ..., fn k)
@(x1, ..., xn) (p) = n k Then:
T(m,p)={v 2 Rn| < v, gradfj(p) >= 0 for j = 1, ..., n k}
Picture 3: tangent vector to M at p
Proof. Consider T1as a vector space spanned by @t@'1(p), ...,@t@'1(p) and T2 = {v 2 Rn| < v, gradfj(p) >= 0 for j = 1, ..., n k}. We want to show: T1 ⇢ TM,p ⇢ T2. Since T1 and T2 are both k-dimensional subspaces ofRnwe have shown then, that necessarily T1= TM,p= T2
and therefore the theorem holds.
(i) (Inclusion T1⇢ TM,p) Set v = 1@'
@t1(p) + ... + k@'
@tk(p) to denote any vector of T1. We define a curve
: ] ✏, ✏[! M ⇢ Rnby (⌧ ) : '(p1+ 1⌧, ..., pk+ k⌧ ) We have (0) = (p) and by chain rule 0(0) = 1@'
@t1(p) + ... +
k@'
@tk(p) = v; this means v2 TM,p
(ii) (Inclusion TM,p ⇢ T2) Consider v 2 TM,p, i.e. v 2 0(0) for a smooth curve : ] ✏, ✏[! M ⇢ Rn with (0) = p. Since this curve proceeds in M it is for j = 1, ..., n k:
fj( (⌧ )) = 0 for|⌧| < ✏1 , (0 < ✏1 ✏) After di↵erentiation we get
0 = Xn i=0
@fj
@xi( (0))d i d⌧ (0) =
=< gradfj(p), 0(0) >=< v, gradfj(p) >
which means v2 T2.
Definition 25. Let M be a k-dimensional manifold with scalar prod- ucth., .i of Rn and p2 M. We call a vector v 2 Rn a normal vector on M in p, if v is perpendicular to TM,p:
hv, wi = 0 for all w 2 TM,p
Normal vectors to M at p build the (n k)- dimensional normal bundle NM,p⇢ Rn. By the above theorem gradf1(p), ..., gradfn k(p) form a basis of NM,p.
3.5 Cotangent space to M at p
In the linear algebra there is the concept of dual vector space. Given a vector space V the dual space V⇤ consists of all linear maps from V toR. We apply this concept on our tangent spaces at every point p on a manifold M and get the corresponding cotangent spaces.
Definition 26. We call TM,p⇤ = Hom(TM,p,R), i.e. the dual space to TM,p, cotangent space to M at p. The map
'⇤: TN,q⇤ ! TM,p⇤
from the cotangent space to N at q to the cotangent space to M at p is called cotangent vector.
Remark 6. Let '⇤ be as above '⇤: TN,q⇤ ! TM,p⇤ and let ' the map between the set of tangent vectors given by ' : TM,p! TN,q. Then
'⇤(!)(p) = !(')(p) holds for p2 TM,pand !2 TN,q⇤ .
3.6 Di↵erential forms
We want to investigate the elements of the just defined spaces. For example, the elements of cotangent space are called 1-forms. After we have considered this most basical case we go on with higher order di↵erential forms.
3.6.1 1-forms
Definition 27. (a) A di↵erential form of order 1 (or a pfaffian form) on an open set U2 Rn is given by the mapping:
! : U ! [
p2U
Tp⇤(U )
where !(p) 2 Tp⇤(U ) for all p 2 U. In other words, this dif- ferential form maps every point p 2 U to a cotangent vector
!(p)2 Tp⇤(U ). We write the value of !(p) on the tangent vector v2 TU,p as < !(p), v >.
(b) Given a smooth function f : U ! R we define its total dif- ferential df as the di↵erential 1-form given by: for p2 U and v2 TU,p:
< df (p), v >:=< gradf (p), v >=
Xn i=1
@f
@xi(p)vi.
The definition works even for a smooth manifold M . Then:
! : M !S
p2MTM,p⇤ . Then we just need to use instead of the standard coordinates inRn the local coordinates of the manifold M .
Representation of 1-forms in coordinates Likewise we constructed by⇣
@
@x1
⌘
p, ...,⇣
@
@xn
⌘
pa basis for the tangent space we want to construct a dual basis for the cotangent space. For this we use the standard coordinate system (x1, ..., xn) of Rn. The process gets the same, if one understands this system to be the local coordinate system of a manifold M .
Consider the di↵erentials dx1, dx2, ..., dxn of a coordinate system (x1, x2, ..., xn) ofRn. The i-th coordinate function is given by
xi:Rn! R; (p1, p2, ..., pn)7! pi
Let ej = (0, ...0, 1, 0, ...0) be the j-th basis vector inRn, where the 1 stands at the j-th place. By definition we have:
< dxi(p), ej>= d
dtxi(p + tej)|t=0= d
dt(pi+ t ij)|t=0= ij
where ij is the Kronecker delta. Therefore we see that the cotan- gent vectors dx1(p), ..., dxn(p) form a basis of TR⇤n,p, which is dual to e1, e2, ..., en. Every cotangent vector '2 TR⇤n,p can be written as
' = Xn
i=0
cidxi(p)
with uniquely determined coefficients ci 2 R. We can conclude that every 1-form ! of an open set U ⇢ Rn can be uniquely written as
! = Xn
i=0
fidxiresp. for all p2 U : !(p) = Xn i=0
fi(p)dxi(p) with functions fi: U ! R.
Lemma 3. Let f : U ! R be a smooth function on open set U ⇢ Rn. Then:
df = Xn
i=0
@f
@xidxi
Proof. We have to show that at every point p2 U the right-hand side and the left-hand side give the same value on every tangent vector v2 TU,p. Since < dxi(p), v >= vi we have:
<X @f
@xi
(p)dxi(p), v >=X @f
@xi
=< df (p), v >
3.6.2 Di↵erential q-forms on M
Definition 28. We call the mapping ! on a manifold M given by
! : M ! [
p2M
⇤q(TM,p⇤ )
a skew-symmetric q-form. The set of all skew-symmetric q-forms at p on M forms a vector space ⇤qTM,p⇤ . Note: ⇤0TM,p⇤ :=R, ⇤1TM,p⇤ = TM,p⇤ .
Representation of a q-form in local coordinates
Again we consider the coordinate system (x1, ..., xn) ofRnand again one can understand those as local coordinates of the manifold M . In canonical coordinate functions x1, x2, ..., xn ofRn a basis of ⇤qTU,p⇤ is given by the elements:
dxi1(p)^ ... ^ dxiq(p), for 1 i1< i2< ... < iq n.
Every di↵erential q-form is representable as:
! : U ! [
p2U
⇤q(TM,p⇤ ),
! = !1^ !2^ ... ^ !q=X
I
↵I(x1, x2, ..., xq)dxi1^ dxi2^ ... ^ dxiq, where each wk, (k = 1, ..., q) is a 1-form, I is an index set and ↵I are unique smooth functions. For all p2 U this means:
!(p) = !1(p)^ !2(p)^ ... ^ !q(p)
=X
I
↵I(p)(x1(p), x2(p), ..., xq(p))dxi1(p)^ dxi2(p)^ ... ^ dxiq(p).
In the following we will derive the theorems and defintions on an open subset U ⇢ Rnwith the standard coordinate system (x1, ..., xn). On a smooth manifold M with local coordinates (x1, ..., xn) all this looks similar, because we provide local charts sending each point of M toRn and which can be glued smoothly together.
Exterior derivative of a di↵erential form For U ⇢ Rnand the q-form ! =P
i1<...<iqfi1...iqdxi1^ ... ^ dxiq
define the (q + 1)-form d! by
d! := X
i1<...<iq
dfi1...iq^ dxi1^ ... ^ dxiq.
Example 5. Take the smooth 1-form ! = Pn
i=1fidxi. Since dfi = Pn
j=1 @fi
@xjdxj we have:
d! = Xn i,j=1
@fi
@xjdxi^ dxj.
Further, dxi^ dxi= 0 and dxi^ dxj = dxj^ dxi and finally:
d! = Xn i<j
✓@fj
@xi
@fi
@xj
◆
dxi^ dxj.
Theorem 5. Let U ⇢ Rn be an open set and !, !1, !2 be smooth q-forms in U and be a smooth r-form, , µ2 R:
(i) d( !1+ µ!2) = d!1+ µd!2
(ii) d(!^ ) = (d!) ^ + ( 1)q!^ (d ) (iii) d(d!) = 0.
Proof. (i) Follows directly from general rules of di↵erentiation.
(ii) Consider first the case q = r = 0, i.e. two smooth functions f, g : U ! R. By the chain rule, @x@i(f g) = @x@
i· g + f · @x@i, we get
d(f g) = gdf + f dg = df^ g + f ^ dg.
Now in the general case we have
! = X
|I|=q
fIdxI, = X
|J|=r
gJdxJ : !^ =X
I,J
fIgJdxI^ dxJ.
We get:
d(!^ ) =X
I,J
(gJdfI+ fIdgJ)^ dxI^ dxJ
=X
I,J
(gJdfI^ dxI^ dxJ+ ( 1)qfIdxI^ dgJ^ dxj)
= (d!)^ + ( 1)q!^ (d ).
(iii) Consider first the case q = 0. For a smooth function f : U! R we have: df =Pn
i=0 @f
@xidxi. By the above example we have:
d(df ) =X
i<j
⇢ @
@xi
✓@f
@xj
◆ @
@xj
✓@f
@xi
◆
= 0
Consider now a twice continuous di↵erentiable q-form
! = X
|I|=q
fIdxI: d! = X
|I|=q
dfI^ dxI.
Since d(dxI) = d(1^ dxI) = 0 follows together with (ii):
d(d!) =X
I
{d(dfI)^ dxI dfI^ d(dxI)} = 0.
Definition 29. Let U 2 Rn be an open set.
(i) A smooth q-form ! in U is called closed, if d! = 0.
(ii) For q 1 we call a smooth q-form ! in U exact, if there exists a smooth (q 1)-form ⌘ in U such that ! = d⌘.
Note: Every exact form is closed, since d d = 0.
3.6.3 Di↵erential n 1-forms on M
We can define di↵erential forms of order n 1 on the open n-dimensional manifold M . We use the following elements as basis
ei(p) = ( 1)i 1(dx1^ ... ^ cdxi^ ... ^ dxn)(p) for 1 i n.
The ’hat’ above dxi means that this factor has to be left out. This allows us to write a (n 1)-dimensional smooth form of the vector space ⇤n 1TM,p⇤ , as
! = Xn
i=1
( 1)i 1fi(dx1^ ... ^ cdxi^ ... ^ dxn) = Xn I=1
fi(p)ei
with smooth coefficients functions fi: M ! R. Since ( 1)i 1
Xn j=1
✓@fi
@xi
dxj
◆
^ dx1^ ... ^ cdxi^ ... ^ dxn=
= @fi
@xidx1^ ... ^ dxi^ ..., ^dxn
we get
d! = Xn i=1
@fi
@dxi
!
dx1^ ... ^ dxn.
We collect the functions fi in a vector field f = (f1, ..., fn) : M ! Rn and define the following n tuple of (n 1)-forms:
d~S := (dS1, ..., dSn), dSi:= ( 1)i 1dx1^ ... ^ cdxi^ ... ^ dxn. We will study d~S closer when we consider integration on manifolds.
We can now determine the n 1-form ! as the scalar product between f and d~S:
! = f· d~S :=
Xn i=1
fi· dSi.
This leads us to the definition:
d! = d(f· d~S) = div(f )dx1^ ... ^ dxn, where
div(f ) = Xn
i=1
@fi
@xi
is the divergence of the vector field f .
Example 6. (Di↵erential 2-forms onR2)
We parametrize the standard coordinates ofR2 with the polar coor- dinates: x = r cos ', y = r sin '.
dx^ dy =
✓@x
@rdr +@x
@'d'
◆
^
✓@y
@rdr + @y
@'d'
◆
=
= (cos 'dr r sin 'd')^ (sin 'dr + r cos 'd') =
= cos ' sin 'dr^ dr + r cos2'dr^ d'+
r sin2'd'^ dr r2sin 'd'^ d' =
= r cos2'dr^ d' + r sin2dr^ d' =
= r(cos2' + sin2')dr^ d' =
= rdr^ d'.
3.7 ⇤ - operation
Problems can often be solved by transition to other coordinates, which are better adjusted to the problem. Change of coordinates from the standard coordinates in Rn to polar coordinates (see the example above) or change of local coordinates from a manifold M to Rn as we have already seen provide good examples. In particular, this means that one maps a point x lying in an arbitrary set U with coordinates (x1, ..., xn) uniquely to another point f (x) = y with the coordinates (y1, ..., yn). Then the set U has under this
transformation f the form f (U ) = V . Functions g on V can then be pulled back to U , by composition g f . Now we want to investigate how di↵erential forms can be pulled back.
3.7.1 Pullback of Di↵erential forms
Let M be a smooth manifold and U⇢ M be a q-dimensional submani- fold. Let V ⇢ Rmbe an open subset with a q-form ! =P
i1<...<iqfi1...iqdxi1^ ...^ dxiq. Furthermore consider a smooth map ':
' = ('1, ..., 'm) : U ! V.
Constructing the pullback of di↵erential forms we should demand that it doesn’t matter if we first perfom di↵erentiation and after that pullback or the other way round. This means, we are looking for a pull back operation '⇤, which commutes with di↵erentiation.
Definition 30. The pullback '⇤! of a di↵erential form ! is defined by:
'⇤! := X
i1<...<iq
(fi1...iq ')d'i1^ ... ^ d'iq.
Let t1, ..., tm denote the canonical coordinate functions inRm. Then define:
d'i= Xm j=1
@'i
@tjdtj.
Example 7. The case q = 1: ! =Pn
i=0fidxi. Then '⇤! =
Xn i=1
Xm j=1
(fi ')@'i
@tjdtj, which means
'⇤! = Xm j=1
gjdtj with gj = Xn
i=1
(fi ')@'i
@tj
. This can be represented in the matrix form as follows. Let
D' = 0 B@
@'1
@t1 ... @'@tm1 ... ...
@'n
@t1 ... @'@tn
m
1 CA
be the Jacobian matrix of ' and both f = (f1, ..., fn) and g = (g1, ..., gm) abbreviated to row vectors, then: g = (f ')D'.
Theorem 6. Let M be a smooth manifold, U ⇢ M be a submanifold, V ⇢ Rm be an open subset and ' : U! V be a smooth map. Further, let !, !1, !2 be q-forms and be a r-form in V and , µ2 R. Then:
(i) '⇤( !1+ µ!2) = '⇤!1+ µ'⇤!2; (ii) '⇤(!^ ) = ('⇤!)^ ('⇤ );
(iii) d('⇤!) = '⇤(d!);
(iv) : W ! U another smooth function on an open set W ⇢ Rp, then: ⇤('⇤!) = (' )⇤!.
Proof. (i) Follows from the linearity of derivatives.
(ii) Set ! =P
IfIdxI, which yields to '⇤! =P
I(fI ')d'I and
=P
JgJdxJ, which yields to '⇤ =P
J(gJ ')d'J. We get the equalities:
'⇤(!^ ) = '⇤ (X
I
fIdxI)^ (X
J
gJdxJ)
!
= '⇤ 0
@X
I,J
fIgJdxI^ dxJ
1 A
=X
I,J
(fIgJ ')dxI^ dxJ
= X
I
(fI ')d'I
!
^ X
J
(gJ ')d'J
!
= ('⇤!)^ ('⇤ ).
(iii) Consider first the case q = 0; i.e. we have a smooth function f : U! R. Using the chain rule we get:
d('⇤f ) = d(f ') = Xm j=1
@(f ')
@tj dtj =
= Xm j=1
Xn i=1
✓ @f
@dxi '
◆@'i
@dtjdtj=
= Xn i=1
✓@f
@xi '
◆ d}'i=
= '⇤ Xn
i=1
@f
@xidxi
!
= '⇤(df ).
For any smooth q-form ! =P
IfIdxI we get '⇤! =X
I
(fI ')d'I,
where we abbreviated d'I:= d'i1^...^d'iqwith I = (i1, ..., iq).
Since the function fI is smooth and 'i is smooth too, follows that the di↵erential form '⇤f is smooth as well. Using (ii) we get:
d('⇤!) =X
d(fI ')^ d'I =X
'⇤(dfI)^ '⇤(dxI) =
= '⇤⇣X
dfI^ dxI⌘
= '⇤(d!).
This proves the most important property: commutativity of dif- ferentiation and pullback.
(iv) Set := ' . For 1 i n we get i= 'i i and with (iii) d i= d('i ) = ⇤(d'i). Further we get:
⇤('⇤!) = ⇤ X
I
(fI ')d'I
!
=
=X
(fI ' ) ⇤(d'I) =
=X
(fI )d I =
= ⇤⇣X fIdxI
⌘= (' )⇤!.
