1. Introduction
One of the great achievements of differential geometry is the Gauss-Bonnet theorem. In its original form, the theorem is a statement about surfaces which connect their geometry in the sense of curvature to the underlying topology of the space, in the sense of the Euler characteristic [1, 2, 3, 4, 5]. The most elementary case of the theorem states that the sum of the angles of a triangle in the plane is π radians. If the surface is deformed, the Euler characteristic does not vary as it is a topological invariant, while the curvature at certain points does change [6, 7, 8]. The theorem states that the total integral of the curvature remains the same, no matter how the deformation is performed. If there is a sphere with a ding, its total curvature is 4π since its Euler characteristic is two. This is the case no matter how big or deep the actual deformation is. A torus has Euler characteristic zero, so its total curvature must also be zero. If the torus carries the usual Riemannian metric from its embedding in ℝ3, then the inside has negative Gaussian curvature, and the outside has positive Gaussian curvature, so the total curvature is zero. It is not possible to specify a Riemannian metric on the torus which has everywhere positive or everywhere negative Gaussian curvature. Manifolds M have dimension n unless stated otherwise [9, 10, 11]. There are many applications of this theorem in both mathematics and mathematical physics such as in gravity [12, 13, 14], string theory [15] and even in the study of Ricci flow [16].
Although the curvature K is defined intrinsically in terms of the metric on the manifold M. It can also be defined for n=2 extrinsically when the metric on M is induced by an embedding M⊂ℝ3. In fact, it ν:M→S2 is the normal map and da is the volume element on S2, then Kdσ=ν∗da so that
∫MKdσ=∫Mν∗da=degν⋅∫S2da=4π⋅degν.E1
Without bringing in differential geometric considerations, it is seen to be the case that degν=1/2χM, where χM is the Euler characteristic of M. Using this fact in (1), the Gauss-Bonnet theorem for a compact oriented surface M, the first version of the theorem is obtained for the case in which the metric on M arises by means of an embedding in ℝ3
∫MKdσ=2πχM.E2
It is the intention here to state and prove a general version of the theorem which applies to manifolds of even dimension, so a surface with n=2 is a special case. An intrinsic proof of the theorem was obtained by Chern 1944. The kind of argument outlined above was used by Hopf in developing the first generalization of the theorem. To outline the basic idea, consider a compact surface Mn⊂ℝn+1 when n is even. If dμg is the volume form on the manifold and dsn denotes the volume element Sn, then
∫MnKndμg=∫Mnν∗dsn=volSn⋅degν=12volSn⋅χMn.E3
This can be extended to any compact oriented Riemannian n-manifold Mng which has even dimension, where Kn in a coordinate system is given by
Kn=12n/2n!∑i1,…,inj1,…,jnRi1i2j1j2⋯Rin−1injn−1jn1gεi1…in⋅1gεj1…jn.E4
The g(4) is the square root of the determinant of the metric. With Kn given by (4), and μg the volume form on the manifold, we are then led to conjecture that
∫MnKndμg=12volSn⋅χMn,E5
where M is a compact, oriented Riemannian manifold with n even.
It is the objective to look at and study some of the ensuing developments which have led to a much deeper understanding of the foundations which underlie this theorem. It will be seen that this development leads to a completely non-computational proof of this deep theorem.
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2. Characteristic classes
When an oriented n-dimensional manifold Mdμg is compact and closed, with dμg is the volume form and μ the orientation of M, so every form has compact support, Stokes theorem leads to the important theorem. Let η be any n−1-form then
∫Mdη=∫∂Mη.E6
Therefore an n-form ω on M, which is not exact, even though it must be closed as all n-forms on M are zero, can be found simply by locating an ω such that
∫Mω≠0.E7
Such a form always exists, as it is known there is a form ω such that for v1,…,vn∈Mp, ωv1…vn>0 if v1…vn=μp. If c:01n→Mμ preserves orientation, c∗ω on 01n is gdx1∧⋯∧dxn for some g>0 on 01n, hence ∫cω>0. This observation leads to this theorem. A smooth, oriented manifold is not smoothly contractible to a point. In fact, it is the shape of M not the size which determines whether or not every closed form on M is exact. More information about the shape of M can be obtained by analyzing more closely the extent to which closed forms are not exact. So how many non-exact n-forms are there on a compact oriented n-manifold If ω is not exact, the same holds for ω+dη for η any n−1-form η. Thus it is necessary to regard ω and ω+dη as equivalent. This suggests an equivalence relation and directs one to think of this in terms of quotient spaces.
For each k, ZkM denotes all closed k-forms on M and it is a vector space. The space BkM of all exact k-forms is a subspace since d2=0. The quotient space is called the k-dimensional de Rham cohomology vector space of M and is defined to be
HkM=ZkM/BkM.E8
The theorem of de Rham states that the vector space is isomorphic to a vector space defined just in terms of the topology of M called the k-dimensional cohomology group of M with real coefficients.
An element of HkM is an equivalence class ω of a closed form ω such that closed forms ω1 and ω2 are equivalent if and only if the difference is exact. In terms of these vector spaces, the Poincaré lemma gives Hkℝn=0, the vector space consisting of just the zero vector if k>0, or HkM=0 if M is contractible and k>0. To compute H0M note B0M=0 as there are no non-zero exact 0-forms as there are no non-zero minus one forms. Thus H0M is the same as the vector space of all C∞ functions f:M→ℝ with df=0. If M is connected, this condition implies f is constant so H0M≡ℝ and its dimension is the number of components of M.
The de Rham cohomolgy vector spaces with compact support HckM are defined similarly to (8), that is, HckM=ZckM/BckM, where ZckM is the vector space of all closed k-forms with compact support and BckM all k-forms dη where η is a k-form with compact support. If M is compact HckM=HkM.
Theorem 2.1. (The Poincaré-Duality Theorem) If M is a connected, oriented n-manfold of finite type, then the map
Π:HkM→Hcn−kME9
is an isomorphism for all k. □
This theorem eventually motivates the introduction of the Euler characteristic for any smooth connected oriented manifold M. Consider then a smooth k-dimensional vector bundle ξ=π:E→M over M. Orientations μ for M, and ν for ξ give an orientation μ⊕ν for the n+k-manifold E, since E is locally a product. Let U1…Ur be a cover of M by geodesically convex sets so small that each bundle ξ restricted to Ui is trivial. Then π−1U1…π−1Ur turns out to be a nice cover for E, so it is a manifold of finite type. For the section and projection maps s,π, π∘s=I on M and s∘π is smoothly homotopic to the identity of E, so the map π∗:HlM→HlE is an isomorphism for all l. The reason for mentioning (6) and Theorem 2.1 is that it shows there is a unique class U∈HckE such that
π∗:μ∪U=μ⊕ν⊂Hcn+kE.E10
This class is called the Thom class of ξ.
A theorem states that if Mμ is a compact oriented, connected manifold ξ=π:E→M an oriented k-plane bundle over M orientation ν, the Thom class U is the unique element of HckE such that for all p∈M, and jp:Fp→E the inclusion map, we have jp∗U=νp. This condition has the implication that ∫Fpνpjpω=1, where U is the class of closed form ω.
The Thom class U of ξ=π:E→M can now be used to determine an element of HkM. Let s:M→E be any section. There is always one, any two are clearly homotopic. Define the Euler class χE⊂HkM of ξ by
χξ=s∗U.E11
If ξ has a non-zero section s:M→E and ω∈CckM represents U, a suitable multiple c⋅s of s takes M to the complement of support ω, so in this case, χξ=c⋅s∗U=0.
The term Euler class is connected with the special case of the bundle TM which has sections which are vector fields on M. If X is a vector field on M having an isolated zero at some point p, Xp=0, but Xq≠0 for q≠p in a neighborhood of p. An index of X at p can be defined. Suppose X is a vector field on an open set U⊂ℝn with an isolated zero at 0∈U. Define fX:U→0→Sn−1 by fXp=Xp/∣Xp∣. If i:Sn−1→U is ip=εp mapping Sn−1 into U, then the map fX∘i:Sn−1→Sn−1 has a certain degree independent of ε for small ε, since maps i1,i2:Sn−1→U correspinding to ε1,ε2 will be smoothly homotopic. This degree is called the index of X at 0. Consider a diffeomorphism h:U→V⊂ℝn with h0=0, so h∗X is the vector field on V such that h∗Xy=h∗Xh−1y. So 0 is an isolated zero of h∗X. It can be shown, if h:U→V⊂ℝn is a diffeomorphism with h0=0 and X has an isolated zero at 0, the index of h∗X at 0 equals the index of X at 0.
As a consequence of this, an index of a vector field on a mainifold can be defined. If X is a vector field on M, with isolated zero at p∈M, choose a coordinate system xU such that xp=p and define the index of X at p to be the index of x∗X at 0.
Theorem 2.2. Let M be a compact, connected manifold with orientation μ, also an orientation for the tangent bundle ξ=π:TM→M. Let X:M→TM be a vector field with only a finite number of zeros and let σ be the sum of indices of X at these zeros. Then
χξ=σ⋅μ∈H0M.E12
□
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3. Pfaffians
An intrinisic expression along with one in a coordinate system for the function Kn on a compact, oriented Riemannian manifold of even dimension has been given already. Another more important way of expressing Kn involves the curvature form Ωji for a positively oriented orthonormal moving frame X1,…,Xn on M. In terms of these forms, the n-form Kndμg, the one to be integrated, can be written down. A sum over permutations such as
∑π∈SnBXπ1…Xπn,E13
can be written just as well as
∑j1,⋯,jnεj1,⋯,jnBXj1…Xjn.E14
Suppose this is the n-fold wedge product
Ωi2i1∧⋯∧Ωinin−1.E15
Since the Ωji are 2-forms, using the definition of wedge product,
Ωi2i1∧⋯∧Ωinin−1X1…Xn=2+⋯+2!2!⋯2!⋅1n!∑j1,⋯jnεi1⋯ipΩi2i1Xj1Xj2⋯Ωinin−1Xjn−1Xjn=12n/2∑j1,⋯jnεj1,⋯,jnRXj1Xj2Xj2Xj1⋯RXjn−1XjnXjnXjn−1=12n/2∑j1,⋯,jnRi1i2j1j2⋯Rin−1injn−1jn.E16
Comparing this to (4), it may be concluded that
Kn=12n/2n!∑i1,⋯,inεi1,⋯,in2n/2Ωi2i1∧⋯∧Ωinin−1X1⋯Xn.E17
When (17) is multiplied by the volume form dμg, it becomes
Kndμg=1n!∑i1,⋯,inΩi2i1∧⋯∧Ωinin−1.E18
By (18) the form on the right does depend on the choice of the positively oriented orthonormal frame, X1,…,Xn. There is a direct way to get this algebraically.
Suppose A is an n×n matrix A=aij with n=2m even. Define the Pfaffian, PfA of A to be
PfA=12mm!∑i1,⋯,inai1i2⋯ain−1in.E19
Note that εi1⋯in does not change when any permutation of the pairs i2l−1i2l. For any set S=h1k1⋯hmkm of pairs of integers between 1 and n, let us define
εS=εh1k1⋯hmkm.E20
It is not necessary to specify any ordering of pairs hiki in S. Also a permutation of the pairs i2l−1i2l does not change the factor ai1i2⋯ain−1in. So for each P above define aS=ah1k1⋯ahmkm. If P is the collection of all such S, we clearly have
PfA=12m∑S∈PεSaS.E21
Theorem 3.1. Let n=2m then for all n×n matrices A and B,
PfBtAB=detB⋅PfA.E22
and Bt denotes the transpose. If B∈SOn then
PfB−1AB=PfA.E23
Proof:
2m⋅m!PfBtAB=∑i1⋯inεi1⋯in∑j1⋯jnbj1i1aj1j2bj2i2⋯bjm−1im−1ajm−1jnbjnin=∑j1⋯jn∑i1⋯inεj1⋯jnbj1i1⋯bjninaj1j2⋯ajn−1jn=∑j1,⋯,jnεj1,⋯,jndetBaj1j2⋯ajn−1jn=2mm!detBPfA.E24
□
This theorem was stated for matrices of real numbers, but PfA can be defined provided the entries of A are in some commutative algebra over ℝ.
Consider again a positively oriented orthogonal moving frame X1,…,Xn on M, with curvature forms Ωji. For each p∈M, the direct sum A=ℝ⊕Ω2Mp⊕Ω2Mp⊕⋯ is a commutative algebra over ℝ under the operation ∧. So one can consider PfΩp, where Ωp is an n×n matrix of connection 2-forms at p
PfΩp=12mm!∑i1,⋯,inεi1,⋯,inΩi1i1∧⋯∧Ωinin−1p.E25
If X′=X⋅a is another positively oriented orthonormal moving frame then ap∈On and the corresponding curvature forms satisfy Ω′=a−1Ωa. Then Theorem 3.1 implies that
PfΩ′p=Pfa−1pΩap=PfΩpE26
so the form ∑i1,⋯,inεi1,⋯,inΩi2i1∧⋯∧Ωinin−1 is well defined.
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4. Bundles of paticular importance
Projective n-space ℙn can be defined as the set of all pairs −pp for p∈Sn⊂ℝn+1 or the set of line through 0 in ℝn+1, since each lines intersects Sn through two anti-podal points. A Grassmannian manifold Gnℝn is the set of all n-dimensional subspaces of ℝN with N>0. Over the Grassmannian manifold GnℝN, there is a natural n-dimensional bundle ζnℝn constructed as follows. The total space of the bundle EζnℝN is the subset of GnℝN×ℝN consisting of all pairs
Ww∈GnℝN×ℝM,w∈W.E27
The projection map which takes EζnℝN→GnℝN is πWw=W. The fibre π−1W over W of GnℝN will be W itself or, more explicitly, Ww:w∈W. A vector space structure is defined on π−1W by using the vector space structure on W⊂ℝN; if a is a scalar, then Ww1+Ww2=Ww1+w2 and aWw=Waw. Also ζnℝN satisfies the local triviality condition.
For M>N there is a natural map α:GnℝN→GnℝM, as an n-dimensional subspace of ℝN can be considered an n-dimensional subspace of ℝM. There is clearly a map α¯:EζnℝN→EζnℝM such that α¯α is a bundle map from ζnℝN to ζnℝM and thus ζnℝN≂α∗ζnℝM.
In algebraic topology, one often considers the union G0ℝ∞ of the increasing sequence Gnℝn+1⊂Gnℝn+1⊂⋯ with weak topology; that is, a set U∈Gnℝ∞=∪lGnℝn+l is open if and only if U∩Gnℝn+l is open in Gnℝn+l for all l. There is a natural n-dimensional bundle ζn over Gnℝ∞ defined in a way similar to ζnℝN such that the following properties are maintained: i for every bundle ξ over a paracompact space X, there is a map f:X→Gnℝ∞ such that ξ≃f∗ζn. ii if f0,f1:X→Gnℝ∞ are maps of a paracompact space X into Gnℝ∞ with f0∗:ζn≃f1∗ζn then f1≃f0.
For this reason ζn is called the universaln-dimensional bundle and Gnℝ∞, is called the classifying space for n-dimensional bundles since equivalence classes of n-dimensional bundles over X are classified by homotopy classes of maps of X into Gnℝ∞. Now Gnℝ∞ is not a manifold so we continue to use the bundles ζnℝN, which are usually called universal bundles.
An orientation for a vector space V is an equivalence class of ordered bases for V where v1…vn∼w1…wn if and only if aij defined by wi=∑jajivj has detaij>0. There are only two such equivalence classes η and −η. An oriented vector space is a pair Vη, where η is an orientation for V.
An orientation for a bundle ξ=π:E→X is a collection η=ηx of orientations for the fibres π−1x which satisfy an obvious compatibility requirement, while an oriented bundle is a pair ξη, where η is an orientation for ξ. Orientation η of ξ gives another −η=−η if X is connected. This is the only other one for ξ. Define ξ1⊕ξ2μ1⊕μ2 to be the sum ξ1⊕ξ2 with the indicated orientation.
Suppose ξ=π:E→M is a smooth oriented n-dimensional vector bundle over a smooth manifold M of any dimension. The Euler class χξ∈HnM was defined by first defining the Thom class Uξ∈HcnE. It can be proved Uξ is the unique class whose restriction to each π−1p is the generator νp=Hcnπ−1p determined by the orientation. This result leads directly into the next theorem.
Theorem 4.1. Let ξ=π:E→M be a smooth manifold where M′ is also a compact manifold. If E is the total space of f∗ξ and f∼:E′→E is a bundle map covering f,
f∼∗Uξ=Uf∗ξ∈HcnE′.E28
Proof: Note f∼ has the property inverse of a compact set is compact, so f∼∗ takes HcnE to HcnE′. Let f∗ξ be π′:E′→M′. If p∈M′ is any point, and ip′:π'−1p→E′ is the inclusion map, then
ip′∗f∼Uξ=f∼∘ip′∗Uξ.E29
Recall how f∗ξ is defined then f∼∘ip′∗Uξ must be the generator of Hcnπ'−1p′, since ifp′∗Uξ is the generator of Hcn(π−1fp′. This shows f∼∗Uξ must be Uf∗ξ. □
The Euler class χξ was defined as s∗Uξ for any section s of ξ. Suppose s=0 is the zero section, which is chosen. It can be shown that
f∗χξ=χf∗ξ∈HnM′.E30
A consequence of (30) is important as it gives the following.
Theorem 4.2. If n is even, then
χQ∼nℝN≠0,N>n.E31
Proof: Since Sn⊂ℝN for N>n, we have a bundle map f∼f:TSn→Eζ∼nℝN so
χTSn=f∗χζ∼nℝN.E32
However, it is known to be the case that χTSn is χSn times the fundamental class of Sn and χSn=2≠0. □
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5. A unique one-form constructed from the curvature
This is an important characteristic class which is important and plays a significant role. Consider principal bundles associated with a smooth oriented n-dimensional vector bundle ξ over a smooth manifold M. There is the principal bundle of frames Fξ of E. If ξ has a Riemannian metric , the bundle OE of orthonormal frames can be considered, which is a principal bundle with group On. Since only paracompact M are considered here, there is an Ehresmann connection ω on OE. Thus ω is a matrix of one-forms ωji on OE with values in on, the curvature form Ω=Dω is a matrix of two-forms Ωji with values in on. A connection ω on OE is equivalent to a covariant derivative operator on E compatible with the metric, and for a general ξ over M, there will be many connections compatible with the metric. One can not be singled out by requiring a symmetric connection which only makes sense for the tangent bundle. As ξ is oriented, we can also consider the bundle SOE of positively oriented frames. If X is connected, it is simply one of the two components of OE, with group SOn and Lie algebra on. A connection ω on SOn again has values in on as does the matrix of two-forms Ω.
If we specialize to the case of a smooth oriented n-dimensional vector bundle ξ=π:E→M over M, with n=2m even. If is a Riemannian metric for ξ and ω is a connection on the corresponding principal bundle ω¯:SOE→M, consider the n-form which is defined on SOE
2m⋅m!PfΩ=∑i1,⋯,inεi1,⋯,inΩi2i1∧⋯∧Ωinin−1.E33
The following is an invariant formulation of a previous theorem.
Theorem 5.1. There is a unique n-form Λ on M such that
ω¯∗Λ=∑i1,⋯,inεi1,⋯,inΩi2i1∧⋯∧Ωinin−1=2mm!PfΩ.E34
Proof: Let X1,⋯,Xn∈MpY1,⋯,Yn∈SOEu be tangent vectors such that πYi→Xi, and choose some u∈ω−1p. Then form Λ must satisfy
ΛX1⋯Xn=2m⋅m!PfΩY1⋯YnE35
This suffices to give uniqueness. If it can be shown this Λ in (35) is well-defined, then existence can be established.
Suppose different tangent vectors Z1,⋯,Zn are taken such that ω¯xZi=Xi. Since ω¯xYi−Zi=0, all Yi−Zi are vertical. However, ΩYZ=0 if either Y or Z is vertical. Consequently,
PfY1⋯Yn=PfΩZ1Y2⋯Yn=PfΩZ1Z2Y3⋯Yn=PfΩZ1⋯Zn.E36
This means the definition of Λ does not depend on the Yi selected. Suppose a different u¯∈ω¯−1p is chosen. Then u¯=RAu=u⋅A for some A∈SOn, and so let Y¯i∈SOEπ be given by Y¯i=RA∗Yi and
PfΩY¯1⋯Y¯n=PfΩRA∗Y1⋯RA∗Yn=PfRA∗ΩY1⋯Yn=PfA−1ΩAY1⋯Yn=PfΩY1⋯Yn.E37
□
Theorem 5.2. The unique n-form Λ in (35) is closed, dΛ=0.
Proof: Suppose X1,⋯,Xn+1∈Mp be given and choose u∈ω¯−1p) and Y1,⋯,Yn+1∈SOEu with ω¯xYi=Xi and hYi the horizontal component of Yi. Then working out dΛ
dΛX1⋯Xn+1=dΛω¯xY1⋯ω¯xYn+1=dΛω¯xhY1⋯ω¯xhYn+1=ω¯∗dΛhY1⋯hYn=dω¯∗ΛhY1⋯hYn+1=2m⋅m!dPfΩhY1⋯hYn+1=2m⋅m!DPfΩY1⋯Yn+1.E38
However, DΩ=0 by Bianchi’s identity and a consequence of this is that (38) vanishes. □
This result applies automatically when ξ is the tangent bundle. The implication of this is that the n-form Λ determines a de Rham cohomolgy class Λ∈HnM of M. The form Λ itself depends on the oriented n-dimensional bundle ξ=π:E→M over M as well as the choice of metric for ξ and connection ω on the corresponding bundle SOE.
Theorem 5.3. The cohomology class Λ is independent of both the metric and the connection ω.
Proof: Suppose two metrics , ′ are given for ξ. Then the corresponding principal bundles SOE and SO′E are equivalent. If f∼:SO′E→SOE is a fiber preserving diffeomorphism which commutes with the action SOn and ω a connection on SOE. Then ω′=f∼∗ω is a connection on SOE. Corresponding curvature forms satisfy Ω′=f∼∗PfΩ so PfΩ′=f∼∗PfΩ. The corresponding forms Λ and Λ′ are in fact equal. It suffices to show any two connection differential forms ω0, ω1 on the same SOE generate forms Λ0, Λ1 whose difference is exact. If π:M×01→M is the projection πpt=p, consider the bundle π∗SOξ over M×01. Induced connections are π∗ω0 and π∗ω1 on π∗SOξ. Let τ:M×01→01 defined here as τpt=t and define a connection
ω=1−τπ∗ω0+τπ∗ω1E39
on π∗SOξ with Ω the connection form. If it maps M to M×01 and is defined as itp=pt, then i0∗ω can be identified with ω0 and i1∗ω with ω1. By Theorems 5.1 and 5.2, which hold for manifolds with and without boundary, there is a closed n-form Λ on M×01 which pulls back to 2mm!PfΩ on the total space of π∗SOξ. A theorem states for any k-form ω on M×01, i1∗ω−i0∗ω=dIω−Idω. So if dω=0, this implies i1∗ω−i0∗ω=dIω. Substituting the form Λ in place of ω into this, it follows that Λ1−Λ0 is exact. □
Thus every smooth oriented smaooth bundle ξ over M of even fibre dimension n determines a de Rham cohomology class Cξ=Λ∈HnM and Cξ=Cη if ξ≃η. It may be asked how does the object Cξ behave with respect to f∗.
Theorem 5.4. Let ξ=π:ξ→M be a smooth oriented bundle over M with fibre dimension n even, let f:M′→M be a smooth map. Then
Cf∗ξ=f∗Cξ∈HnM′.E40
Proof: The total space of f∗ξ is called E′. Let f∼:E'→E be the bundle map covering f. If is a metric on E, then f∼∗ is a metric on E. There is an equivalence f¯:SOE′→SOE covering f with ω¯' taking SOE′ to M′ and ω¯ mapping SOE to M.
If ω is a connection on SOE, then f¯∗ω is a connection on SOE′. It is seen that the corresponding connection forms satisfy Ω′=f∼∗Ω. Aa a result, we have
PfΩ′=Pff¯∗Ω=f¯∗PfΩ.E41
For n-forms Λ on M given by Theorem 5.1, we then have
ω¯'∗f∗Λ=f¯∗ω¯∗Λ=2m⋅m!f¯∗PfΩ=2m⋅m!PfΩ′.E42
This means f∗Λ must be the n-form Λ′ on M′ given in (31). □
When ξ is a smooth oriented bundle of odd fibre dimension, the definition of C may be extended. It would be remarkable if it were the case that Cξ were always a constant multiple of χξ. To this end, the following theorem is needed.
Theorem 5.5. Let ξi=πi:Ei→M for i=1,2 be smooth oriented vector bundles over M of fibre dimension n1 and n2. If ni=2mi, then
Cξ1⊕ξ2=m1+m2!m1!m2!Cξ1∪Cξ2.E43
If n1 or n2 is odd, this reduces to Cξ1⊕ξ2=0.
Proof: Pick two metrics which are Riemannian for each ξi and set ⋅⋅=⋅⋅1⊕⋅⋅2 on ξ1⊕ξ2=π:E→M. Let ω¯i:SOEi→M and ω¯:SOE→M be the corresponding principal bundles. Over M consider the product principal bundle Q=SOE1∗SOE2 with corresponding group SOn1×SOn2⊂SOn1+n2 whose fiber over p∈M is the direct product ω¯1−1×ω¯2−1p, so this bundle is a subset of SOE.
Let ρi be the projection maps for Q which project this down onto either of its factors. If ωi are connections on SOEi, with curvature forms Ωi, then
ρ1∗ω1⊕ρ2∗ω2=ρ1∗ω100ρ2∗ω2E44
is a connection on Q and the curvature form is
Ω=ρ1∗Ω1⊕ρ2∗Ω2=ρ1∗Ω100ρ2∗Ω2E45
The connection ω¯ can be extended uniquely to a connection ω∼ on SOE. The requirement ω∼σM=M determines ω∼ at the new vertical vectors, hence ω∼ is determined at all points of Q, and then at all points of SOE by the requirement ω∼RA∗Y=AdA−1ω∼Y.
At any point e∈Q, the horizontal vectors for ω∼ are the same as that for ω¯. At E, it holds that Ω¯=Ω∼ for tangent vectors to Q which implies, using PfA⊕B=PfA⋅PfB, that
PfΩ∼=PfΩ¯=Pfρ1∗Ω1∧Pfρ2∗Ω2=ρ1∗PfΩ1∧ρ2∗Ω2.E46
Consequently, if Λi are the forms given by (34), then at e it must hold that on tangent vectors to Q
ω¯∗Λ=2m1+m2m1+m2!PfΩ∼=m1+m2!m1!m2!2m1m1!ρ1∗PfΩ1∧2m2m2!ρ2∗PfΩ2=m1+m2!m1!m2!ρ1∗ω¯1∗Λ1∧ρ2∗ω¯2∗Λ2=m1+m2!m1!m2!ω¯1∗Λ1∧ω¯2∗Λ2.E47
This implies that
Λ=m1+m2!m1!m2!Λ1∧Λ2.E48
□
Corollary 5.1. If the oriented bundle ξ=π:E→M has a nowhere zero section s, then
Cξ=0.E49
Proof: Let E1⊂E be written
∪p∈Mℝ⋅sp,E50
and let E2⊂E be the orthogonal complement
∪p∈Mℝ⋅sp⊥E51
with respect to some Riemannian metric on E. Then ξ1=π1∣E1:E1→M is an oriented one-dimensional bundle. Consequently, ξ2=π2∣E2:E2→M is also an oriented bundle since ξ is oriented. Clearly ξ≃ξ1⊕ξ2. An application of the previous result shows that Cξ=0. □
This theorem is almost enough to characterize χ as we can now show the statement which relates Cξ and the Euler class.
Corollary 5.2. If ξ=π:E→M is a smooth vector bundle of fibre dimension n over a compact oriented manifold M, then the class Cξ∈HnM is a multiple of the Euler class χξ.
Proof: Suppose S is the sphere bundle S=e∈E:ee=1, which is constructed with respect to some Riemannian metric on E. Let π0:S→X be the restriction π∣S. The bundle π0∗ξ has a nowhere zero section. Corrollary 5.1 and Theorem 5.4 then yield
π0∗Cξ=Cπ0∗ξ=0.E52
However, there is a theorem which states a class α∈HnM satisfies π0∗α=0 if and only if α is a multiple of χξ. It can now be inferred that Cξ is a multiple of the Euler class χξ. □
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6. The Gauss-Bonnet-Chern theorem
If Corollary 5.2 is applied to the tangent bundle of a compact oriented manifold M of dimension n which is even, the class CTM∈HnM is some multiple of the Euler class χTM. This fact is not so interesting because HnM is one-dimensional since it means CTM=0 if χTM=0. The corollary does lead to something interesting when applied to the universal bundle.
Theorem 6.1 For every even n, there is a constant βn such that
Cξ=βnχξ.E53
for all smooth oriented n-dimensional bundles ξ over compact oriented manifolds. In this sense, it is universal.
Proof: Begin with the bundles z∼etanℝN for N>n. Corollary 5.2 implies there are constants βn,N such that
Cζ∼nℝN=βn,Nχζ∼nℝN∈Hnζ∼nℝN.E54
If j:G∼nℝN→G∼nℝM is the natural inclusion, then j∗ζ∼NℝN≃ζ∼nℝN. Equation (30) and Theorem 5.4 yield
Cζ∼nℝN=j∗Cζ∼nℝM=j∗χζ∼nℝM.E55
Thus, (54), (55) give
βn,Nχζ∼nℝN=βn,Mχζ∼nℝN.E56
Since χζ∼nℝN≠0 by Theorem 4.2, this implies that βn,N=βn,M for all M,N>1. This common number is called βn, and we have
Cζ∼nℝN=βnχζ∼nℝN.E57
However it is known that any smooth oriented n-dimensional bundle ξ over a compact manifold M is equivalent to f∗ζ∼nℝN for some smooth map f:M→G∼ℝN, then
Cξ=Cf∗ζ∼nℝN=f∗Cζ∼nℝN=βnf∗χζ∼nℝN=βnχξ.E58
□
The constant βN is universal in nature and it may be asked whether it can be computed. Since it has this universality property, it suffices to compute this constant for a special case where the calculation is easier and in turn implies another application of the next theorem.
Theorem 6.2 For integer n=2m, the constant βn in Theorem 6.1 is
βn=n!2VSn=πm2nm!.E59
If M is a compact manifold of even dimension n=2m then
∫MKndμg=πm2nm!n!χM.E60
Proof: Let ξ be the tangent bundle TM of a compact oriented manifold of dimension n. Now (17) gives a formula for Kndμg, where Ωji are curvature forms for some positively oriented orthonormal moving frame. This implies that the form Λ in Theorem 5.1 for the bundle SOξ=SOTM is
Λ=n!Kndμg.E61
If κ is the fundamental class of M then
∫MKndμgκ=1n!∫MΛκ=1n!Cξ=βnn!χξ=βnn!χM⋅κ.E62
Hence, equating the coefficients of κ on both sides,
∫MKndμg=βnn!χM.E63
Consider the case of a specific manifold M=Sn in (63), where Kn=1 so the left side of (63) reduces to VSn
VSn=βnn!χSn=2βnn!.E64
Since the volume VSn is known to be πm2n+1m!/n!, solve (64) for βn in terms of VSn ,
βn=n!2VSn=πm2nm!.E65
This value of βn can be put back into (60) and for the manifold M, it follows that
∫MKndμg=πm2nm!n!χM.E66
□
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7. The theorem for manifolds with boundary
Theorem 5.5 played a large part in the proof of (64). It allowed us to state that if ξ=π:E→M is an oriented n-dimensional vector bundle with sphere bundle π0:S0→M, then π0∗Cξ=0 was a large part of the proof of (). If Λ is the n-form on M representing Cξ, then the n-form π0∗Λ on S is exact π0∗Λ=dΦ for some n−1-form on S. Suppose ξ=TM∗ and let X be a unit vector bundle on M which has an isolated singularity at p∈M Let Bε be a closed ball of radius ε around p and set Mε=M\intBε where int denotes the interior. Then XMε is a manifold with boundary, the image of Mε under the section X:M\p→S. Consequently
∫MΛ=∫M−pΛ=limε→0∫MεΛ=limε→0∫MεX∗π0∗Λ=limε→0∫X(Mεπ0∗Λ
=limε→0∫XMεdΦ=limε→0∫∂XMεΦ.E67
If indXp is the index of X at p
∫MΛ=indXp∫π0−1pΦ=χM∫π0−1pΦ.E68
Since n=2m we also have the Gauss-Bonnet-Chern Theorem 6.2
∫MΛ=∫Mn!Kndμg=πmm!2nχM,E69
we finally obtain
∫π0−1pΦ=πmm!2n.E70
Let M∂M be a compact orientable manifold with boundary with Euler characteristic χM=dimH0M−dimH1M+⋯. A compact oriented manifold M2 can be constructed, the double of M, by taking two disjoint copies of M and identifying corresponding points of ∂M.
Theorem 7.1 The Euler characteristic of the manifold M2 is given by
χM2=2χM−χ∂M.E71
Proof: Let U and V be open neighborhoods of the two copies of M in M2 such that HkU≡HV≡HkM for all k and HkU∩V≡Hk∂M for all k. So there is the sequence 0→H0M2→⋯→HkM2→HkU⊕HkV→⋯→HkU∩V→Hk+1M2. When the sequence is exact, a theorem can be applied to obtain the result. □
This is very interesting since it claims different things depending on whether the dimension n of M is even or odd. When n is odd χM2=0 hence χM=1/2χ∂M which implies χ∂M must be even. But when n is even, χ∂M=0, so the previous theorem implies
2χM=χM2.E72
Corollary 7.1 Let M be a compact orientable manifold with boundary of even dimension n. Let X be a vector field on M with only finitely many zeros all in M\∂M such that X is outward pointing on ∂M. The sum of indices of X is χM.
Proof: Modify X near ∂M so it is an outward pointing unit normal ν on the boundary and so there are no new zeros. Then there is a vector field on M2 which looks like X on one copy of M and −X on the other. Since n is even, the index −X of an isolated zero is the same as the index of X at that zero. The Theorem of Poincaré-Hopf on the sum of indices of X gives twice the sum of the indices of X equals χM2=2χM by (72). □
Theorem 7.2 Let M be a compact oriented Riemannian manifold with boundary of even dimension n=2m, tangent bundle π:TM→M and associated sphere bundle π0=π∣S:S→M. Let ω be a connection on the principal bundle ω¯:SOTM→M, with curvature form Ω. Let Λ be the unique n-form on M with
ω¯∗Λ=∑εi1,⋯,inΩi2i1∧⋯∧Ωinin−1=2mm!PfΩ,E73
and Φ an n−1-form on S with π0∗Λ=dΦ. Let ν:∂M→S be the outward pointing unit normal on ∂M. Then
∫MKndμg=1n!∫MΛ=πmm!2nn!χM+1n!∫∂Mν∗Φ.E74
Proof: Extend ν to a vector field X on M with only finitely many zeros p1,…,pk∈M\∂M. Let Biε be the closed balls of radius ε centered at pi which are disjoint from each other and from ∂M. Put Mε=M\∪i=1kintBiε. Now integrate the form Λ over M and use (70)
∫MΛ=limε→0∫∂XMεΦ=∫ν∂MΦ+∑i=1klimε→0∫∂BiεΦ=∫∂Mν∗Φ+πmm!2n∑i=1MindXpi=∫∂Mν∗Φ+πmm!2nχM.E75
The last line makes use of Corollary 7.1. □
Theorem 7.2 presents one way in which Theorem 6.2 can be generalized to the case of manifolds with boundary. At this point an interpretation for the first term in (75) is not easy to provide. It is required to obtain an explicit Φ such that π0∗Λ=dΦ. In fact such a Φ can be constructed.