A generalization of the Gauss-Bonnet and Hopf-Poincaré theorems
AA GENERALIZATION OF THE GAUSS-BONNET AND HOPF-POINCAR ´ETHEOREMS
F. A. ARIAS AND M. MALAKHALTSEV
Abstract.
We consider a locally trivial fiber bundle π : E → M over a compact oriented two-dimensional manifold M , and a section s of this bundle defined over M \ Σ, where Σ is a discretesubset of M . We call the set Σ the set of singularities of the section s : M \ Σ → E .We assume that the behavior of the section s at the singularities is controlled in the following way: s ( M \ Σ) coincides with the interior part of a surface S ⊂ E with boundary ∂S , and ∂S is π − (Σ).For such sections s we define an index of s at a point of Σ, which generalizes in the natural way theindex of zero of a vector field, and then prove that the sum of this indices at the points of Σ can beexpressed as integral over S of a 2-form constructed via a connection in E .Then we show that the classical Hopf-Poincar´e-Gauss-Bonnet formula is a partial case of our result,and consider some other applications. G -structure1. Introduction
Let M be a 2-dimensional compact oriented manifold, and V be a vector field on M . The classicalHopf-Poincar´e theorem affirms that the sum of indices of zeros of V is equal to the Euler characteristicof M :(1) (cid:88) z ∈ Z ( V ) ind z ( V ) = χ ( M ) , where Z ( V ) is the set of zeros of V and ind z ( V ) is the index of V at z (see for example [1]). If, inaddition, the manifold M is endowed by a Riemannian metric g , then the Gauss-Bonnet theoremsays that the Euler characteristic is expressed in terms of the curvature K of this metric:(2) χ ( M ) = 12 π (cid:90) M Kdσ, where dσ is the area form of g . Therefore, we have that(3) (cid:88) z ∈ Z ( V ) ind z ( V ) = 12 π (cid:90) M Kdσ,
Then this formula can be generalized for sections of a unit sphere subbundle of the tangent bundle ofa Riemannian manifold, thus we arrive at the Euler class (see, e. g. [2], Appendix 20: Gauss-Bonnettheorem) and it also can be generalized to arbitrary sphere bundles [1].In this paper we find a generalization of this formula to arbitrary locally trivial bundles withcompact fibers over two-dimensional closed manifolds. Namely, we consider a locally trivial fiberbundle π : E → M over a compact oriented two-dimensional manifold M , and a section s of this a r X i v : . [ m a t h . DG ] O c t F. A. ARIAS AND M. MALAKHALTSEV bundle defined over M \ Σ, where Σ is a discrete subset of M . We call the set Σ the set of singularitiesof the section s : M \ Σ → E .We assume that the behavior of the section s at the singularities is controlled in the following way: s ( M \ Σ) coincides with the interior part of a surface S ⊂ E with boundary ∂S , and ∂S is π − (Σ)(see details in subsection 2.1). It is worth noting that this idea was used by S.-S.Chern in [3]. Forsuch sections s we define an index of s at a point of Σ, which generalizes in the natural way theindex of zero of a vector field (subsection 2.2), and then prove that the sum of these indices at thepoints of Σ can be expressed as integral over S of a 2-form constructed via a connection in E . Thusin Theorem 2 we obtain a generalization of the formula (3).In Section 4 we show that the classical fact that the index of a vector field on a closed two-dimensional Riemann manifold is the integral of the curvature of the Levi-Civita connection can beobtained from our generalized version. Also in this section we consider the projective bundles andprojective connections and find a relation between the index of a section of a projective bundle andthe curvature of a projective connection in this bundle.In Section 5 our results are applied to the theory of G -structures with singularities. If ¯ P → M is a¯ G -principal bundle, which is reduced to a G -principal bundle over M \ Σ, where Σ is a submanifoldof M , we say that ¯ P is a G -principal bundle with singular set Σ. If the ¯ G -principal bundle ¯ P isa subbundle of the linear frame bundle L ( M ) of a manifold M , we say that ¯ P is a G -structurewith singularities. Almost all singularities which appear in classical differential geometry define G -structures with singularities. Some examples are: a vector field on a Riemannian n -dimensionalmanifold defines SO ( n − n -dimensional manifold M which degenerates along a submanifold, determines a SO ( n )-structure with singularities on M (examples of such metrics were considered for example in [4]), 3-webs with singularities on a two-dimensional manifolds, which in particular appear in considerations of algebraic ordinary differentialequations on manifolds (see, e. g., [5]) define a G -structure with singularities, where G ⊂ GL (2) is theLie subgroup of diagonal matrices [6], also sub-Riemannian manifolds with singularities determinethe corresponding G -structure with singularities [7].A G -principal bundle ¯ P with singularities determines a section s : M \ Σ → ¯ P /G , which is asection with singularities of the bundle ¯
P /G . For the case dim M = 2, we apply the results obtainedin Section 2 to this situation and obtain the relation between the index of this section and theconnection in the bundle ¯ P → M (see Section 5 for details).2. Singularity of section and its index
Sections with singularities. Resolution of singularity.
Let M be a 2-dimensional orientedclosed manifold, and ξ = ( π : E → M ) a locally trivial fiber bundle with typical fiber F and structuregroup G . We will assume that G is a connected Lie group. For each x ∈ M , by F x we will denotethe fiber of ξ over x , and by i x : F x → E the inclusion. Let Σ ⊂ M be a discrete subset. Let usconsider a section s : M \ Σ → E , we will call Σ the singular set of the section s .Assume that there exist a -dimensional oriented compact manifold S with boundary ∂S and amap q : S → E such that the map q restricted to ◦ S = S \ ∂S is a diffeomorphism between S \ ∂S and s ( M \ Σ). We will call the map q a resolution of the singularities of the section s . Theorem 1. a) π ◦ q : S \ ∂S → M \ Σ is a diffeomorphism. b) The map π ◦ q : S → M is surjective. c) π ◦ q ( ∂S ) = Σ . GENERALIZATION OF THE GAUSS-BONNET AND HOPF-POINCAR´E THEOREMS 3 d) Let ∂S = (cid:83) i =1 ,k ∂S i , where ∂S i are the boundary components. Then for any x ∈ Σ , there existone and only one boundary component ∂S i such that π − ( x ) = q ( ∂S i ) . e) For any x ∈ Σ exists a neighborhood U of x such that (1) U is diffeomorphic to a disk; (2) There exists a closed set V ⊂ S diffeomorphic to a ring such that the boundary of V consistsof two connected components C and C (cid:48) , where the connected component C is a closed curve in S \ ∂S and the other connected component C (cid:48) is the connected component ∂S i of the boundary ∂S which corresponds to x by d), and (4) π ◦ q ( C ) = ∂U, π ◦ q ( V \ ∂V ) = U \ x, π ◦ q ( ∂S i ) = x Remark . In what follows we will label each point x ∈ Σ such that π − ( x ) = q ( ∂S i ) by the corre-sponding index i . Proof. a) s : M \ Σ → s ( M \ Σ) is a diffeomorphism and π | s ( M \ Σ) : s ( M \ Σ) → M \ Σ is thediffeomorphism inverse to s . Therefore π ◦ q : S \ ∂S → M \ Σ is a diffeomorphismb) For each point x ∈ M \ Σ, the point y = q − ( s ( x )) ∈ S has the property that π ◦ q ( y ) = x . Letus take a point x ∈ Σ and a sequence of points { x n ∈ M \ Σ } converging to x . The sequence of points y n = q − ( s ( x n )) ∈ S has a convergent subsequence { y n k } because S is compact. If y = lim n k →∞ y n k ,then(5) π ◦ q ( y ) = lim n k →∞ πq ( y n k ) = lim n k →∞ x n k = x. c) As π ◦ q is surjective and π ◦ q ( S \ ∂S ) = M \ Σ because q ( S \ ∂S ) = s ( M \ Σ), we have that π ◦ q ( ∂S ) = Σ.d) and e) For a point x ∈ Σ let us take a neighborhood U diffeomorphic to an open disk such that U = U \ { x } ⊂ M \ Σ. The closure U = Γ (cid:116) U (cid:116) { x } , where Γ is the boundary of U .Let V = q − s ( U ). Then π ◦ q ( V ) = U because M and S are compact. As π ◦ q : S \ ∂S → M \ Σis a diffeomorphism, and Γ ⊂ M \ Σ, we have a curve C ⊂ ( V \ V ) ∩ ( S \ ∂S ) such that π ◦ q ( C ) = Γ.This means that C is a part of the boundary ∂V which lies in the interior part of S . Now, by b)we know that π − ( x ) ⊂ (cid:116) i ∈ I ∂S i , I ⊂ { , · · · , n } . Finally we have that V = C (cid:116) V (cid:116) ( (cid:116) i ∈ I ∂S i ),and we know that V is diffeomorphic to U , and C ⊂ S \ ∂S is diffeomorphic to Γ, that is C isdiffeomorphic to a circle S . Thus V is an open set in S diffeomorphic to an open ring, and theboundary of V consists of a closed curve in the interior part of S and some components of theboundary of S . However, this means that the boundary of V contains one and only one component ∂S i of ∂S , otherwise the fundamental group of π ( V ) (cid:54)∼ = Z . Thus we get the affirmation d) and thedecomposition V = V = C (cid:116) V (cid:116) ∂S i with the properties given by (4). Therefore, we have provede). (cid:3) Remark . From the proof of items d) and e) of Theorem 1 it follows that, for each boundarycomponent ∂S i , we have a closed neighborhood V i of ∂S i and a diffeomorphism of manifolds withboundary u i : S × [0 , → V i such that u i ( S × { } = ∂S i , U = π ◦ q ◦ u i ( S × [0 , x i ∈ Σ diffeomorphic to a disk, π ◦ q ◦ u i ( S × { } is the boundary of U , and π ◦ q ◦ u i ( S × { } = x (see Figure 1). Example . Let ξ be a unit vector bundle of R , π : R × S → R , and consider a section s : R → E which takes value s ( ρ, ϕ ) = (cos nϕ, sin nϕ ) with respect to the polar coordinate system on R \ (0 , F. A. ARIAS AND M. MALAKHALTSEV
Figure 1.
The surface S The point (0 ,
0) is the singular point of this section. If we take the coordinate system ( ρ, ϕ, ψ ) on π − ( R \ (0 , s ( ρ, ϕ ) = ( ρ, ϕ, nϕ ). Now the map u : S × [0 , → E is given with respect tocoordinates ( x, y, ψ ) on E = R × S by u ( α, t ) = ((1 − t ) cos α, (1 − t ) sin α, nα ). Therefore u ( S ×{ } )is a curve consisting of helices, u ( S × { } ) is the fiber (0 , × S ⊂ E represents the image of u . Figure 2.
Example 12.2.
Index of a point x ∈ Σ . First let us first observe that π : E → M is trivial on the homotopylevel. Let ( U, ψ : π − ( U ) → U × F ) be a chart of the atlas of ξ . Let(6) η = p F ◦ ψ : π − ( U ) → F, where p F : U × F → F is the canonical projection onto F . For each x ∈ U the map η restrictedto F x = π − ( x ) induces a diffeomorphism η x : F x → F . Note that if we take another chart ( U (cid:48) , ψ (cid:48) : π − ( U (cid:48) ) → U (cid:48) × F ), and η (cid:48) : π − ( U (cid:48) ) → F is the corresponding map, then on π − ( U (cid:84) U (cid:48) ) we have GENERALIZATION OF THE GAUSS-BONNET AND HOPF-POINCAR´E THEOREMS 5
Figure 3.
Example 1that(7) ψ (cid:48) ◦ ψ − : ( U ∩ U (cid:48) ) × F → ( U ∩ U (cid:48) ) × F, ( x, y ) (cid:55)→ ( x, g ( x ) y ) , where g : U ∩ U (cid:48) → G is the gluing map of the charts. Now, for any x ∈ U ∩ U (cid:48) , we have η (cid:48) x ◦ η − x ( y ) = g ( x ) y , and, as G is connected, η (cid:48) x ◦ η − x : F → F is homotopic to the identity map. Thismeans that for any x ∈ M we have well defined isomorphisms of the homotopy and (co)homologygroups:(8) π ∗ ( η x ) : π ∗ ( F x ) → π ∗ ( F ) ,H ∗ ( η x ) : H ∗ ( F x ) → H ∗ ( F ) , H ∗ ( η x ) : H ∗ ( F ) → H ∗ ( F x ) , which do not depend on the chart. In particular, the following diagram is commutative:(9) π ( F x ) × H ( F x ) (cid:104)· , ·(cid:105) (cid:39) (cid:39) π ( η x ) × H ( η − x ) (cid:47) (cid:47) π ( F ) × H ( F ) (cid:104)· , ·(cid:105) (cid:120) (cid:120) R where (cid:104)· , ·(cid:105) is the natural pairing through the integration. This means that, for [ γ ] ∈ π ( F x ) and[ α ] ∈ H ( F ), we have(10) (cid:10) [ γ ] , H ( η x )[ α ] (cid:11) = (cid:90) γ η ∗ x α = (cid:90) η x γ α = (cid:104) π ( η x )[ γ ] , [ α ] (cid:105) , and this equality neither depends on the chart, nor on the choice of representatives γ or α of theirequivalence classes. Definition 1.
For x i ∈ Σ the map q ◦ u i : S × [0 , → E restricted to S × { } gives a map γ i : S → F x i , and therefore an element ind x i ( s ) ∈ π ( F ) via the isomorphism π ( η x ) (see (8)). Thiselement ind x i ( s ) will be called the index of the section s at the point x i ∈ Σ. Remark . The index does not depend on the choice of the maps u i . F. A. ARIAS AND M. MALAKHALTSEV
Example . The index of the section s at the point (0 ,
0) in Example 1 is equal to nγ , where γ is agenerator of the group π ( S ). Definition 2.
For a ∈ H ( F ) we define the index ind x i ( s ; a ) of the section s at the point x i ∈ Σ with respect to a as(11) ind x i ( s ; a ) = (cid:104) a, ind x i ( s ) (cid:105) = (cid:90) γ α = (cid:90) S γ ∗ α, where γ : S → F is the curve representing the element ind x i ( s ) ∈ π ( F ) and α ∈ Ω ( F ) is a closedform representing the element a ∈ H ( F ). The index of a section s with respect to a is the number(12) ind ( s ; a ) = (cid:88) x i ∈ Σ ind x i ( s ; a ) . Remark . By (11),(13) ind x i ( s ; a ) = (cid:90) γ i η ∗ x α, where [ α ] = a and γ i is defined in Definition 1. Example . Let a = [ dφ ] ∈ H ( S ; R ), where dφ is the angular form on S , then a is a base of H ( S ; R ). Then index of the section s at the point (0 ,
0) in Example 1 with respect to a is equal to n . 3. Connection and the Gauss-Bonnet theorem
Let H be a connection in the fiber bundle E . Let us denote by V the vertical distribution , that isthe distribution of the tangent spaces to the fibers, then we have T E = H ⊕ V .3.1. Vertical cohomology class.Statement 1.
Let a ∈ H ( F ) and H be a connection in E . There exists a -form α ∈ Ω ( E ) suchthat (1) α | H = 0 ; (2) for each x ∈ M , di ∗ x α = 0 and [ i ∗ x α ] = H ( η x ) a .Proof. Let us take an atlas ( U i , ψ i ) of the bundle ξ , and fix a 1-form ¯ α ∈ Ω ( F ) such that [ ¯ α ] = a . Let ρ i be the partition of unity subordinate to the covering { U i } , and η i : π − ( U i ) → F (see (6)). Thentake the collection of 1-forms ¯ α i = η ∗ i ¯ α ∈ Ω ( π − ( U i )), and then construct (cid:101) α = (cid:80) i π ∗ ρ i ¯ α i ∈ Ω ( V ).Then(14) i ∗ x (cid:101) α = (cid:88) i ρ i ( x ) i ∗ x ¯ α i = (cid:88) i ρ i ( x ) i ∗ x η ∗ i ¯ α ∈ Ω ( F x ) . Therefore,(15) di ∗ x (cid:101) α = (cid:88) i ρ i ( x ) i ∗ x η ∗ i d ¯ α = 0and(16) [ i ∗ x (cid:101) α ] = (cid:88) i ρ i ( x ) [ i ∗ x η ∗ i ¯ α ] = a, GENERALIZATION OF THE GAUSS-BONNET AND HOPF-POINCAR´E THEOREMS 7 because (cid:80) i ρ i ( x ) = 1. We have the decomposition T E = H ⊕ V , and therefore we can define thedesired α ∈ Ω ( E ) by α | H = 0 and α | V = (cid:101) α . (cid:3) Linear connection in the vertical bundle V → E . Let ∆ be a distribution on a manifold B . A partial ∆ -connection in a vector bundle π : E → B is given by a covariant derivative: ∇ : ∆ × Γ( π ) → Γ( π )such that(17) ∇ λX + Y s = λ ∇ X s + ∇ Y s, λ ∈ F ( B ) , X ∈ X (∆) , s ∈ Γ( π ); ∇ X s + s = ∇ X s + ∇ Y s ; X ∈ X (∆) , s , s ∈ Γ( π ); ∇ X f s = Xf s + f ∇ X s ; f ∈ F ( B ) , X ∈ X (∆) , s ∈ Γ( π ) . Linear connection in the vector bundle V → E induced by the connection H . . The connection H induces a partial linear H -connection ∇ H in the vector bundle V → E :(18) ∇ HX Y = ω ([ X, Y ]) , X ∈ X h ( E ) , Y ∈ X v ( E )in the following way. One can easily check that the properties (17) hold true. The geometricalsense of the partial H -connection is as follows. The connection H induces a parallel translationof the fibers of the bundle E along the paths on the base M : for γ : [0 , → M , we have thediffeomorphism Π γ : F γ (0) → F γ (1) . Therefore, if γ h is a horizontal lift of γ , we have the paralleltranslation Π Hγ = d Π | γ h (0) : V γ h (0) E → V γ h (1) E along the curve γ h . As Π H is linear, from this paralleltranslation we get a partial linear connection ∇ H which is exactly the connection (18).3.2.2. Linear connection in the vector bundle V → E . Let us fix a linear partial V -connection ∇ V inthe bundle V → E , then the connection ∇ V induces a liner connection in each fiber. We will assumethat this connection does not have torsion. Then let us consider the linear connection ∇ = ∇ H + ∇ V in the bundle V → E :(19) ∇ X Y = ∇ HX h Y + ∇ VX v Y, where X = X h + X v corresponds to the decomposition T E = H ⊕ V , and Y ∈ X v ( E ).3.3. Structure equations of connection.Statement 2.
Let ξ = ( π : E → B ) be a vector bundle of rank r over a manifold B , and ∇ be alinear connection in this bundle. Then we can extend the exterior differential d : Ω q ( B ) → Ω q +1 ( B ) to a differential operator D : Ω q ( B ) ⊗ Γ( ξ ) → Ω q +1 ( B ) ⊗ Γ( ξ ) , where Γ( ξ ) is the module of sectionsof ξ so that (20) D ( ω ⊗ s ) = dω ⊗ s + ( − | ω | ω ∧ ∇ s. Proof.
Let { e α } α =1 ,r be a local frame field of ξ over U ⊂ B . Then any ω ∈ Ω q ( B ) ⊗ Γ( ξ ) restricted to U can be written in the form ω = ω α e α . If we change the frame field to { e α (cid:48) } α (cid:48) =1 ,r , then e α (cid:48) = A αα (cid:48) e α F. A. ARIAS AND M. MALAKHALTSEV and ω α (cid:48) = A α (cid:48) α ω α , where A αα (cid:48) are functions on U . Then,(21) D ( ω α (cid:48) e α (cid:48) ) = dω α (cid:48) e α (cid:48) + ( − q ω α (cid:48) ∧ ∇ e α (cid:48) = d ( A α (cid:48) β ω β ) A αα (cid:48) e α + ( − q ω α (cid:48) ∧ ∇ ( A αα (cid:48) e α )= dA α (cid:48) β ∧ ω β A αα (cid:48) e α + A α (cid:48) β dω β A αα (cid:48) e α + ( − q ω α (cid:48) ∧ dA αα (cid:48) e α + ( − q ω α (cid:48) ∧ A αα (cid:48) ∇ e α = ( A αα (cid:48) dA α (cid:48) β ∧ ω β + ( − q ω β ∧ A α (cid:48) β dA αα (cid:48) ) e α + dω α e α + ( − q ω α ∧ ∇ e α = ( A αα (cid:48) dA α (cid:48) β + A α (cid:48) β dA αα (cid:48) ) ∧ ω β e α + D ( ω α e α ) = D ( ω α e α ) , because A αα (cid:48) dA α (cid:48) β + A α (cid:48) β dA αα (cid:48) = d ( A αα (cid:48) A α (cid:48) β ) = d δ αβ = 0. (cid:3) Remark . D ( ω ⊗ s ) = ω ∧ R ( s ), where R is the curvature tensor of the linear connection ∇ . Statement 3.
For ω ∈ Ω ( B ) ⊗ Γ( E ) , and X, Y ∈ X ( B ) , (22) Dω ( X, Y ) = ∇ X ( ω ( Y )) − ∇ Y ( ω ( X )) − ω ([ X, Y ]) . Proof.
Let { e α } be a local frame field of ξ , and ω = ω α e α . Then(23) ∇ X ( ω ( Y )) = ∇ X ( ω α ( Y ) e α ) = X ( ω α ( Y )) e α + ω α ( Y ) ∇ X e α , therefore(24) ∇ X ( ω ( Y )) − ∇ Y ( ω ( X )) − ω ([ X, Y ]) = X ( ω α ( Y )) e α + ω α ( Y ) ∇ X e α − Y ( ω α ( X )) e α − ω α ( X ) ∇ Y e α − ω α ([ X, Y ]) e α = dω α ( X, Y ) e α − ω α ∧ ∇ e α ( X, Y ) = Dω ( X, Y ) . (cid:3) Now let us assume that we have chosen the linear connection ∇ in the vector bundle V → E given by (19), and therefore we have the corresponding differential operator D : Ω q ( B ) ⊗ X V ( E ) → Ω q +1 ( B ) ⊗ X V ( E ), where X V ( E ) is the Lie algebra of vertical vector fields. Statement 4 (Structure equations) . Let ω : T E → V be the connection form of the connection H .Then (25) Dω = Ω , where Ω( X, Y ) = Dω ( X h , Y h ) ∈ Ω ( E ) ⊗ X V ( E ) is horizontal.Proof. It is sufficient to prove that Dω ( X, Y ) = 0 if Y ∈ X v ( E ). We will apply (22). If X ∈ X h ( E ),then(26) Dω ( X, Y ) = ∇ X ω ( Y ) − ∇ Y ω ( X ) − ω ([ X, Y ]) = ∇ X Y − ω ([ X, Y ]) = 0by the definition of ∇ (see (18) and (19)). If X ∈ X v ( E ), then(27) Dω ( X, Y ) = ∇ X ω ( Y ) − ∇ Y ω ( X ) − ω ([ X, Y ]) = 0again by the definition of ∇ and because the connection ∇ V is torsion free. (cid:3) The tensor Ω ∈ Ω ( E ) ⊗ X V ( E ) is called the curvature form of the connection H . Corollary 1.
The curvature form Ω vanishes if and only if H is integrable. GENERALIZATION OF THE GAUSS-BONNET AND HOPF-POINCAR´E THEOREMS 9
Proof.
Indeed, by Statement 3, we have that, for any horizontal vector fields X and Y ,(28) Dω ( X, Y ) = − ω ([ X, Y ]) , therefore Dω ( X h , Y h ) = 0, for any X h and Y h if and only if the Lie bracket of horizontal vectorfields is again a horizontal vector field, this is equivalent to integrability of H . (cid:3) Structure equations of the -form α . The decomposition
T E = H ⊕ V induces the bigradingon the space of differential forms on E : Ω k ( E ) = ⊕ l + m = k Ω ( l,m ) ( E ), and d : Ω (0 ,l ) ( E ) → Ω (0 ,l +1) ( E )because the distribution V is integrable.Let a ∈ H F and (cid:101) α ∈ Ω ( V ) is a 1-form such that η ∗ ([ (cid:101) α | F x ]) = a . In Statement 1 we haveconstructed a 1-form α ∈ Ω (0 , E ) such that dα vanishes on each fiber. Note that the form α ( X ) = (cid:101) α ( ω ( X )) satisfies the conditions (1) and (2) of Statement 1.Let us calculate dα . As X (cid:101) α ( ω ( Y )) = ∇ X (cid:101) α ( ω ( Y )) + (cid:101) α ( ∇ X ( ω ( Y )), we have(29) dα ( X, Y ) = Xα ( Y ) − Y α ( X ) − α ([ X, Y ]) = X (cid:101) α ( ω ( Y )) − Y (cid:101) α ( ω ( X )) − (cid:101) α ( ω ([ X, Y ]) = ∇ X (cid:101) α ( ω ( Y )) + (cid:101) α ( ∇ X ( ω ( Y )) − ∇ Y (cid:101) α ( ω ( X )) + (cid:101) α ( ∇ Y ( ω ( X )) − (cid:101) α ( ω ([ X, Y ]) = ∇ X (cid:101) α ( ω ( Y )) − ∇ Y (cid:101) α ( ω ( X )) + (cid:101) α ( Dω ) . Here we use Statement 3. Let us consider the 2-form(30) P ∈ Ω ( E ) , P ( X, Y ) = ∇ X (cid:101) α ( ω ( Y )) − ∇ Y (cid:101) α ( ω ( X ))If X, Y ∈ X h ( E ), then P ( X, Y ) = 0. Now if
X, Y ∈ X v ( E ), then P ( X, Y ) = d (cid:101) α ( X, Y ) = 0.Therefore, the form P ∈ Ω (1 , ( E ), and for X ∈ X h ( E ) and Y ∈ X v ( E ), by (30), we get that(31) P ( X, Y ) = ∇ X (cid:101) α ( ω ( Y )) = X ( (cid:101) α ( Y )) − (cid:101) α ( ∇ X Y ) = Xα ( Y ) − α ([ X, Y ]) = L X α. Now, using Statement 3, we arrive at the following statement.
Statement 5.
The form α constructed in Statement 1 lies in Ω (0 , ( E ) and dα = θ (1 , + θ (2 , , where θ (1 , ∈ Ω (1 , and θ (2 , ∈ Ω (2 , , and (32) θ (1 , ( X, Y ) = ( L X α )( Y ) , θ (2 , = (cid:101) α (Ω) . Note that the forms θ (1 , and θ (2 , in (32) depends only on the connection H and the form α . Nowwe apply the Stokes theorem to the surface S and arrive at a generalization of the Gauss-Bonnettheorem: Theorem 2.
For a ∈ H ( F ) , let α ∈ Ω ( E ) be the corresponding vertical form constructed inStatement 1. Also let dα = θ (1 , + θ (2 , as in Statement 5. Then (33) (cid:90) S q ∗ θ (1 , + q ∗ α (Ω) = (cid:88) i =1 ,k index x i ( s ; a ) . Proof.
By the Stokes theorem,(34) (cid:90) S dq ∗ α = (cid:90) ∂S q ∗ α = k (cid:88) i =1 (cid:90) ∂S i q ∗ α. We have that(35) (cid:90) ∂S i q ∗ α = (cid:90) S ×{ } u ∗ i q ∗ α = (cid:90) S γ ∗ i α = (cid:90) γ i α = ind x i ( s ; a ) , Therefore,(36) k (cid:88) i =1 (cid:90) ∂S i q ∗ α = ind x i ( s ; a ) . On the other hand, by Statement 5,(37) (cid:90) S dq ∗ α = (cid:90) S q ∗ dα = (cid:90) S q ∗ θ (1 , + q ∗ θ (2 , . and thus we get the required statement. (cid:3) Examples
Classical Hopf theorem.
Let (
M, g ) be a compact oriented two-dimensional Riemannianmanifold, and E = T ( M, g ) the bundle of unit vectors. Then E is a local trivial bundle withstructure group SO (2) and typical fiber S . Any V ∈ X ( M ) defines a section s = (cid:107) V (cid:107) V : M \ Σ → E ,where Σ is the set of zeros of V . Assume that the set Σ is discrete. We will consider trivializations ofthe bundle E given by local orthonormal frames { e , e } . For each x i ∈ Σ let us consider a orientationpreserving diffeomorphism c i : ( D, → ( U i , x i ), where D ⊂ R is a disk centered at the origin 0. Let { e a } be an orthonormal frame field over U i which determines a diffeomorphism π − ( U i ) ∼ = U i × S .Now let consider a map(38) t i : S × [0 , → U i , ( φ, ρ ) (cid:55)→ c i ((1 − ρ ) cos φ, (1 − ρ ) sin φ )) , and let ψ i : U i \ { x i } → S be the local representation of the section s with respect to the giventrivialization, this means that(39) s = cos ψ i e + sin ψ i e . Now t i induces a diffeomorphism S × [0 , → U i \ { x i } and therefore we get a map ψ i ◦ t − i : S × [0 , → S . Now assume that the map ψ i ◦ t − i can be prolonged to a map Ψ i : S × [0 , → S .Under this assumption we can construct the desired compact 2-dimensional manifold S and the map q : S → E with properties (1) and (2) (see Section 2). Remark . In geometric terms, our assumption means that the angle ψ of the unit vector field s withthe vector field e (see (39)) has limit when we approach to the point x i by some curve, but the limitdepends on the curve.In order to construct S , first we take S (cid:48) = M \ ∪ ki =1 U i , this is a submanifold with boundarywhich consists of the curves Γ i = ∂U i = c i ( ∂D ), then consider the disjoint union S (cid:48)(cid:48) of k copies ofthe cylinder S × [0 , ∼ = ∂D × [0 , S (cid:48)(cid:48) to S (cid:48) by the map C : (cid:116) ki =1 ∂D × → (cid:116) ki =1 Γ i determined by the maps(40) c i | ∂D : S × { } ∼ = ∂D × { } → Γ i . Thus we get the manifold S = S (cid:48) ∪ C S (cid:48)(cid:48) . Now the map q : S → E is defined as follows: for each p ∈ S (cid:48) , we set q ( p ) = s ( p ), and for each p ∈ S (cid:48)(cid:48) = (cid:116) ki =1 ∂D × [0 , p belongs to i -th copy of S × [0 , ∼ = ∂D × [0 , q ( p ) = Ψ i ( p ). From the construction it is clear that q is well defined GENERALIZATION OF THE GAUSS-BONNET AND HOPF-POINCAR´E THEOREMS 11 on S (cid:48) ∩ S (cid:48)(cid:48) = (cid:116) i =1 ,k Γ i . Let a ∈ H ( S ) be the class represented by the angle form θ . The Riemannianconnection ∇ of the metric g defines a connection H in the bundle E → M . Let α ∈ Ω ( E ) be the1-form constructed in Statement 1. In terms of the trivialization(41) U × S → π − ( U ) , ( q, ψ ) (cid:55)→ cos ψe ( q ) + sin ψe ( q ) . where U ⊂ M and { e , e } is an orthonormal frame field, the form α is written as follows:(42) α = θ + π ∗ G, where ∇ e = Ge and ∇ e = − Ge , and G = G e + G e , G kij are the connection coefficients of G with respect to the orthonormal frame { e , e } . Remark . If W = cos ψe + sin ψe , then ∇ W = ( dψ + G )( − sin ψe + cos ψe ) = ( dψ + G ) W ⊥ ,therefore W ( t ) = ( γ ( t ) , ψ ( t )) is parallel if and only if ddt ψ + G ( ddt γ ) = 0. This means that, in termsof the trivialization, the connection form is exactly the form α given by (42). Therefore α vanishesover H (condition (1) in Statement 1). At the same α restricted to a fiber is θ , this means satisfiescondition (2) in Statement 1.Denote by Ψ ai : S → S the map Ψ i : S × [0 , → S restricted to S × { a } . Then Ψ i is homotopicto Ψ i , therefore deg Ψ i = deg Ψ i . However, Ψ i = ψ i ◦ t i restricted to S × { } and its degreecoincides with the degree of ψ i restricted to Γ i = ∂U i , therefore the deg Ψ i is equal to the classicalindex Ind x i ( s ) of the vector field s at the point x i . Therefore,(43) ind x i ( s, a ) = (cid:90) γ i α = (cid:90) S (Ψ i ) ∗ θ = deg Ψ i = deg Ψ i = Ind x i ( s ) . This means that the classical index
Ind x i ( s ) of vector field s at the singular point x i coincides withthe index ind x i ( s, a ). On the other hand dα = π ∗ dG = π ∗ Ke ∧ e , where K is the curvature of themetric g . Therefore, in Statement 5 θ (1 , = 0 and α (Ω) = π ∗ ( Ke ∧ e ). Thus, (43) gives us(44) (cid:90) S q ∗ π ∗ ( Ke ∧ e ) = (cid:88) i =1 ,k Ind x i ( s ) . At the same time, by the construction, π ◦ q : S \ ∂S → M \ Σ is a diffeomorphism, Σ is a set ofmeasure zero, and Ke ∧ e is a 2-form defined globally on M , therefore(45) (cid:90) S q ∗ π ∗ ( Ke ∧ e ) = (cid:90) M Ke ∧ e and we get the classical result(46) (cid:90) M Ke ∧ e = (cid:88) i =1 ,k Ind x i ( s ) . Example: Projective bundles.
Projective bundles.
Definition 3. A projective bundle is a locally trivial bundle Q → M with typical fiber R P n andstructure group P GL ( n ). The number n is called the rank of Q . Example . Let E → M be a vector bundle. Then the bundle P ( E ) of 1-dimensional subspaces ofthe fibers of E is called the projectivization of E and it is projective bundle. Projective bundle as projectivization of a vector bundle.
If we have a projective bundle Q → M ,then can we find a vector bundle E such that Q = P ( E )? Note that for A ∈ GL ( n + 1), we havedet( λA ) = λ n +1 det A . Now for an even n , the group P GL ( n ) = GL ( n + 1) /A ∼ λA , λ ∈ R , isisomorphic to SL ( n +1) because for each equivalence class [ A ] ∈ P GL ( n ) one can find a representative B = (det A ) − / ( n +1) A ∈ [ A ] such that det B = 1. This B is unique because for B, B (cid:48) ∈ [ A ] such thatdet B = det B (cid:48) = 1, we have that B (cid:48) = λB and λ n +1 = 1, this implies that λ = 1. Therefore in thiscase P GL ( n ) ∼ = SL ( n + 1). For an odd n , the group P GL ( n ) = P GL + ( n ) (cid:116) P GL − ( n ), where(47) P GL + ( n ) = { [ A ] | det A > } , P GL − ( n ) = { [ A ] | det A < } . By the same arguments as for case of even n we get that P GL + ( n ) ∼ = SL ( n ) / ± I . Note that P GL − ( n )is diffeomorphic to P GL + ( n ). If the structure group P GL ( n ) of a projective bundle Q reduces to P GL + ( n ), then Q is called orientable , on the contrary Q is called no orientable . Example . If a vector bundle E is no orientable, then P ( E ) is no orientable.Now let consider the following situation. Let (cid:101) G be a Lie group, Z = Z ( (cid:101) G ) be the center of (cid:101) G , and G = (cid:101) G/Z . So we have the exact sequence(48) 0 (cid:47) (cid:47) Z i (cid:47) (cid:47) (cid:101) G p (cid:47) (cid:47) G (cid:47) (cid:47) . Let P → M be a G -principal bundle, U = { U i } be a good covering of a manifold M , and g ij : U ij → M be the transition functions of M . As the covering U is good, we can lift g ij up to (cid:101) g ij : U ij → (cid:101) G of g ij , this means that the following diagram is commutative(49) (cid:101) G p (cid:15) (cid:15) U ij (cid:101) g ij (cid:63) (cid:63) g ij (cid:47) (cid:47) G Consider the Cech cochain z = (cid:110) z ijk : U ijk → (cid:101) G (cid:111) over the covering U , where(50) z ijk = (cid:101) g ij (cid:101) g jk (cid:101) g ki . As p ( z ijk ) = g ij g jk g ki = 1, the functions z ijk take values in Z , therefore in fact z ∈ ˇ C ( U ; Z ). Statement 6. (1)
The ˇCech cochain z is a normalized cocycle. (2) The ˇCech cohomology class [ z ] ∈ ˇ H ( M ; Z ) is trivial if and only if there exists a (cid:101) G -principalbundle (cid:101) P such that P ∼ = (cid:101) P /Z .Proof.
Let us prove that { z ijk } is a ˇChech cocycle on U . First of all note that we have(51) (cid:101) g ij (cid:101) g jk = z ijk (cid:101) g ik = (cid:101) g ik z ijk . Then(52) ( (cid:101) g ij (cid:101) g jk ) (cid:101) g ki = z ijk (cid:101) g ik (cid:101) g ki = z ijk and (cid:101) g ij ( (cid:101) g jk (cid:101) g ki ) = (cid:101) g ij (cid:101) g ji z jki = z jki , therefore z ijk = z kij = z jki . At the same time,(53) z − ijk (cid:101) g − ik = ( z ijk (cid:101) g ik ) − = ( (cid:101) g ij (cid:101) g jk ) − = (cid:101) g kj (cid:101) g ji = z kji (cid:101) g ki , GENERALIZATION OF THE GAUSS-BONNET AND HOPF-POINCAR´E THEOREMS 13 hence follows that z − ijk = z kji = z jik . Moreover z ijk = 1 if at least two indices coincide. Thus z = { z ijk } is a normalized cochain. Now we have(54) ( (cid:101) g ij (cid:101) g jk )( (cid:101) g kj (cid:101) g jl ) = z ijk (cid:101) g ik z kjl (cid:101) g kl = z ijk z kjl z ikl g il and(55) ( (cid:101) g ij (cid:101) g jk )( (cid:101) g kj (cid:101) g jl ) = ( (cid:101) g ij ( (cid:101) g jk ( (cid:101) g kj ) (cid:101) g jl = (cid:101) g ij (cid:101) g jl = z ijl (cid:101) g il , thus z ijk z kjl z ikl = z ijl , this implies(56) δz = z jkl z − ikl z ijl z − ijk = 0 . If z = δu , where u = { u ij ∈ Z } , then(57) z ijk = u ij u jk u ki . Therefore we can construct the (cid:101) G -principal bundle (cid:101) P → M with the transition functions (cid:101) g (cid:48) ij = (cid:101) g ij u ji because(58) (cid:101) g (cid:48) ij (cid:101) g (cid:48) jk (cid:101) g (cid:48) ki = (cid:101) g ij u ji (cid:101) g jk u kj (cid:101) g ki u ik = (cid:101) g ij (cid:101) g jk (cid:101) g ki u ji u kj u ik = z ijk u ji u kj u ik = 1 . Now we have that p ( (cid:101) g (cid:48) ij ) = g ij and from the following lemma it follows that there exists a principalbundle morphism π : (cid:101) P → P written with respect to the trivializations as ( x, (cid:101) g ) → ( x, p ( (cid:101) g )). Lemma 1.
Let G a , a = 1 , be Lie groups. Let π a : P a → M , a = 1 , be a G a -principal bundle and f : G → G be a Lie group homomorphism. Assume that U = { U i } is a trivializing covering for P and P at the same time. If the transition functions g aij with respect to the covering U are related by f ( g ij ) = g ij , then there exists a principal fiber bundle homomorphism F : P → P corresponding to f , i. e. F ( p g ) = F ( p ) f ( g ) .Proof. We define F i : π − ( U i ) → π − ( U i ) by the commutative diagram(59) π − ( U i ) F i (cid:47) (cid:47) ψ i (cid:15) (cid:15) π − ( U i ) ψ i (cid:15) (cid:15) U i × G × f (cid:47) (cid:47) U i × G where ψ a are the trivializations which are principal bundle isomorphisms, therefore ψ a ( p a g a ) = ψ a ( p a ) g a , where the action of G a on U i × G a is given by ( x, g a ) g (cid:48) a = ( x, g a g (cid:48) a ). From this follows that F i ( p g ) = F i ( p ) g . On the other hand, for p ∈ π − ( U ij ) we have F i ( p ) = F j ( p ). Indeed, we havethe commutative diagram for a = 1 , U ij × G a ( x,g a ) → ( x,g aji g a ) (cid:47) (cid:47) U ij × G a π − ( U ij ) ψ ai (cid:102) (cid:102) ψ aj (cid:56) (cid:56) Let p ∈ π − ( U ij ) and ψ i ( p ) = ( x, g ), then ψ j ( p ) = ( x, g ji g ). Also ψ i F i ( p ) = ( x, f ( g )), hence ψ j F i ( p ) = ( x, g ji f ( g )). Therefore,(61) ψ j F j ( p ) = ( x, f ( g ji g )) = ( x, f ( g ji ) f ( g )) = ( x, g ji f ( g )) = ψ j F i ( p ) , thus F j = F i on π − ( U ij ) and therefore { F i } determine a homomorphism F : P → P . (cid:3) From the construction of the bundle (cid:101) P it follows that P = (cid:101) P /Z and the principal fiber bundlehomomorphism constructed in Lemma 2 is exactly the projection (cid:101) P → P = (cid:101) P /Z . (cid:3) Let Q be a projective bundle of rank n , n is odd. Assume that Q is orientable. Let { U i } be agood covering of M , U i ...i k = U i (cid:84) · · · (cid:84) U i k , and g ij : U ij → P GL ( n ) the corresponding cocycle oftransition functions. As U ij are contractible, we can choose the maps ˆ g ij : U ij → SL ( n + 1) suchthat p (ˆ g ij ) = g ij , where p : SL ( n + 1) → P GL + ( n ) is the canonical projection. Then p (ˆ g ij ˆ g jk ˆ g ki ) = g ij g jk g ki = id ∈ P GL ( n ), therefore e ijk = ˆ g ij ˆ g jk ˆ g ki takes values in Z = {± I } . Note that Z = {± I } is the center of the group SL ( n + 1). Corollary 2. If Q → M is a projective bundle of odd rank n and is oriented, then the obstructionto existence of a vector bundle E such that Q = P ( E ) is an element in H ( M, Z ) . Definition 4.
Let Q → M be a projective bundle of rank n . The principal bundle P GL ( Q ) withthe group P GL ( n ) adjoint to Q is called the bundle of projective frames . Example . On the total space of a vector bundle E → M the group R ∗ = R \ { } acts freely byscaling, and Q = P ( E ) is the quotient of E with respect to this action. Therefore, we have the R ∗ -principal bundle π : E → Q . Also we have the induced action of R ∗ on the total space of the linearframe bundle GL ( E ), and the quotient space is P GL ( Q ). The quotient map GL ( E ) → P GL ( Q )is R ∗ -principal bundle. If E is an orientable vector bundle of rank n + 1, where n is odd, then thestructure group GL ( n + 1) of GL ( E ) reduces to SL ( n + 1), and in this case we have the doublecovering SL ( E ) → P GL ( Q ). For an orientable vector bundle E of rank n + 1, one can take a metric g on E , therefore the GL ( n + 1)-principal bundle GL ( E ) of linear frames reduces to a SO ( n + 1)-principal bundle of positively oriented orthonormal frames SO ( E ). Therefore, for Q = P ( E ), thebundle P GL ( Q ) reduces to the subgroup SO ( n + 1) / ± I ⊂ P GL ( n ) = SL ( n + 1) / ± I .4.2.3. Projective connections.
Let Q → M be a projective bundle. A projective connection is aconnection in P GL ( Q ). Any linear connection in E induces a projective connection in P ( E ). Indeed,the parallel transport preserves one-dimensional subspaces in fibers of E . At the same time thecanonical projection GL ( E ) → P GL ( P ( E )) is a principal bundle morphism corresponding to themorphism of groups GL ( n +1) → P GL ( n ). Therefore, any connection in GL ( E ) determines a uniqueconnection in P GL ( E ) [8](Chapter II, Proposition 1). Statement 7.
Two linear connections ∇ and ∇ (cid:48) in the vector bundle E induce the same projectiveconnection in P ( E ) if and only if the deformation tensor T = ∇ (cid:48) − ∇ has the form (62) T iab = ξ i δ ab . If Q = P ( E ), where E is an oriented vector bundle, then for any projective connection in Q we can find an equi-affine connection ∇ in E with respect to a given volume form θ ∈ Ω n +1 E ,i. e. ∇ θ = 0, such that ∇ induces the given connection in Q . Indeed in this case SL ( E, θ ) covers
P GL ( Q ) = SL ( E, θ ) / Z , Z = ± I , and any connection in P GL ( P ( E )) lifts to a connection in SL ( E, θ ). Statement 8.
Let E be an oriented vector bundle, and θ ∈ Ω n +1 E be a volume form in E . Thenthere is one-to-one correspondence between equi-affine connections on E with respect to θ and theprojective connections in P ( E ) . GENERALIZATION OF THE GAUSS-BONNET AND HOPF-POINCAR´E THEOREMS 15
Remark . In terms of parallel translation this correspondence can be explained as follows. Let Γ bea projective connection in Q = P ( E ) induced by a linear connection ∇ in E . In terms of the paralleltranslation this means that a section L of Q along γ is parallel with respect to the connection Γ ifthere exists a section s of E along γ which is parallel with respect to ∇ and generates L . This canbe expressed also in terms of covariant derivative: L = [ s ] is parallel with respect to Γ if and only if ∇ ˙ γ s = µs . Now any projective frame field L = { L , L , · · · , L n } in Q = P ( E ) determines a framefield E = { E , · · · , E n } in E such that L i = [ E i ], and L = [ E ], where E = E + E + · · · + E n ,up to a scaling. If in addition we have a volume form θ , the projective frame field determines thelinear frame field { E i } up to a sign. Then ∇ ˙ γ E i = µ i E i and ∇ ˙ γ E = µ E , hence follows that µ = µ = · · · = µ n . If we assume that the linear connection ∇ is equi-affine with respect to avolume form θ , then ∇ θ = 0 and the function µ = ddt log θ ( E , · · · , E n ).4.3. Section of projective bundle and reduction of the bundle of projective frames.Statement 9.
Let Q be a projective bundle of rank . Let Af f (1) ⊂ P GL (1) be the subgroup ofaffine transformations of line. Then Q ∼ = P GL ( Q ) /Af f (1) and a section of projective bundle Q isequivalent to the reduction of the bundle P GL ( Q ) to the subgroup Af f (1) .Proof.
This follows from the fact that the group
P GL (1) acts transitively on R P , and the isotropysubgroup of a point in R P is the group Af f (1). (cid:3)
The structure equations of the form α . The vertical form α . Let E → M be an oriented vector bundle and rankE = 2. Let Q = P ( E )be the corresponding projective bundle. Also let us take a projective connection Γ in Q . The typicalfiber of Q is R P . Let us consider the canonical projection j : R \ { } → R P , j ( v , v ) = [ v : v ],and the corresponding double covering ¯ : S ι (cid:44) → R \ { } j → R P . Let us consider the form(63) θ = 12 1( v ) + ( v ) (cid:0) − v dv + v dv (cid:1) ∈ Ω ( R \ { } ) , Then ι ∗ θ is the angular form on S and there exists ¯ α ∈ Ω ( R P ) such that j ∗ ¯ α = θ . Then, as (cid:82) S ι ∗ θ = 2 π , we have that (cid:82) R P ¯ α = π . Therefore, ¯ α is a generator of H ( R P ). Now let us takea projective connection H is Q which is determined by an equi-affine connection ˆ H in E . Nowwe can construct the form α in Ω ( Q ) such that α | H = 0, di x α = 0, and i ∗ x α represents the class[ ¯ α ] ∈ H ( R P ). Let us take a metric g in E , the volume form θ associated with g , and let ω be theconnection form on E , that is the projection onto the vertical subspace parallel to ˆ H . Now let usconsider the 1-form(64) ˆ α v ( X ) = 12 g ( v, v ) θ ( ω ( X ) , v ) ∈ Ω ( E ) , where v ∈ E , E = E \ M ), and 0 is the zero section of E . This form ˆ α vanishes on H , and themetric g allows us to choose the trivialization of the bundle E such that the maps i x are isomorphismsof the euclidean vector spaces ( E x , g x ) and R n with the standard metric. Therefore by i x the formˆ α restricted to the fibers is sent to the form (63). Now let us consider local coordinate system( x i , y a ) on E such that g = ( y ) + ( y ) , then θ = − y dy + y dy . Consider the local coframe dx i , Dy a = dy a + Γ iab y b dx i . Lemma 2. (65) d ( Dy a ) = Γ iab Dy b ∧ dx i + R ijab y b dx i ∧ dx j , where R ijab = ∂ i Γ jab − ∂ j Γ iab + Γ iac Γ jcb − Γ jac Γ icb is the curvature tensor of the linear connection in E .Proof. We have(66) d ( Dy a ) = d ( dy a + Γ iab y b dx i ) = d Γ iab y b ∧ dx i + Γ iab dy b ∧ dx i = ∂ j Γ iab y b dx j ∧ dx i + Γ iab ( Dy b − Γ jbc y c dx j ) ∧ dx i =Γ iab Dy b ∧ dx i + (cid:0) ∂ j Γ iab y b dx j ∧ dx i − Γ iab Γ jbc y c dx j ∧ dx i (cid:1) =Γ iab Dy b ∧ dx i + (cid:0) ∂ j Γ iab y b dx j ∧ dx i + Γ jab Γ ibc y c dx j ∧ dx i (cid:1) =Γ iab Dy b ∧ dx i + R ijab y b dx i ∧ dx j . (cid:3) The form ˆ α is written with respect to this frame as follows:(67) ˆ α = 12 1( y ) + ( y ) (cid:0) − y Dy + y Dy (cid:1) . Now let us assume that the connection is adapted to the metric g in E , in this case Γ iab = − Γ iba .Then we have(68) d (( y ) + ( y ) ) = 2 (cid:0) y Dy + y Dy (cid:1) , because(69) d (( y ) + ( y ) ) = 2 (cid:0) y dy + y dy (cid:1) =2 (cid:0) y ( Dy − Γ i y dx i ) + y ( Dy − Γ i y dx i ) (cid:1) = 2 (cid:0) y Dy + y Dy (cid:1) . Also we have that(70) dy ∧ Dy = Dy ∧ Dy − Γ i y dx i ∧ Dy ,y dDy = Γ i y Dy ∧ dx i + R ij ( y ) dx i ∧ dx j ,dy ∧ Dy = Dy ∧ Dy − Γ i y dx i ∧ Dy ,y dDy = Γ i y Dy ∧ dx i + R ij ( y ) dx i ∧ dx j , Therefore,(71) d ( y Dy − y Dy ) = (cid:0) dy ∧ Dy + y dDy − dy ∧ Dy − y dDy (cid:1) =2 Dy ∧ Dy + R ij (cid:0) ( y ) + ( y ) (cid:1) dx i ∧ dx j . GENERALIZATION OF THE GAUSS-BONNET AND HOPF-POINCAR´E THEOREMS 17
Thus,(72) d ˆ α =12 (cid:20) d (cid:18) y ) + ( y ) (cid:19) ∧ (cid:0) − y Dy + y Dy (cid:1) + 1( y ) + ( y ) d (cid:0) − y Dy + y Dy (cid:1)(cid:21) = − y ) + ( y ) ) (cid:0) y Dy + y Dy (cid:1) ∧ (cid:0) − y Dy + y Dy (cid:1) ++ 1( y ) + ( y ) Dy ∧ Dy + 12 R ij dx i ∧ dx j = 12 R ij dx i ∧ dx j . Note that Ω = π ∗ ( R ij dx i ∧ dx j ) is the curvature form of the connection H restricted to π − ( U ),where U is the coordinate neighborhood. Theorem 3.
Let E → M be a vector bundle, Q = P ( E ) , and j : P ( E ) → Q . (1) There exist a -form α ∈ Ω ( Q ) such that ˆ α = j ∗ α . (2) α | H = 0(3) di x α = 0 , and i ∗ x α represents the class [ ¯ α ] ∈ H ( R P ) . (4) dα = π ∗ (Ω) . Corollary 3.
Let E → M be a vector bundle and Q = P ( E ) . Assume that g is a metric in E .Let Ω be the curvature form of the connection in Q induced by a connection in E adapted to g . Let a ∈ H ( R P ) is represented by the form ¯ α . If s : M \ Σ → Q is a section with singularities, (73) ind ( s ; a ) = (cid:90) M Ω . Gauss-Bonnet Theorem in Principal G -bundles with Singularities G -structures with singularities.Definition 5. Let ¯ G be a Lie group, and G ⊂ ¯ G be a Lie subgroup of G . Let M be a manifold,and Σ be a submanifold of M . A principal ¯ G -bundle ¯ P ( M, ¯ G ) is called a principal G -bundle withsingularities Σ if the structure group ¯ G of ¯ P | M \ Σ reduces to G . Example . Let X be a vector field on a manifold M with a discrete set of zeros Σ ⊂ M . Thenthe frame bundle L ( M ) is a principal GL ( n − M . It is clear that on M \ Σ the bundle L ( M ) reduces to the principal GL ( n − X at the corresponding point. Example . If M is a two-dimensional oriented Riemannian manifold, and X is a vector field on M with a discrete set of zeros Σ ⊂ M , then the orthonormal frame bundle SO ( M ) is a principal { e } -bundle with singularities where the set of singularities Σ because the vector field induced trivializationof SO ( M ) over M \ Σ. Example . Let M be a Riemannian manifold of dimension n . The principal GL ( n )-bundle of frames L ( M ) is a principal O ( n )-bundle with an empty set of singularities. Let us recall that there is an important relation between H -reductions of a principal G -bundle P → M and sections of the fiber bundle P/H → M which is given by the following statement. Statement 10 ([8](Chapter I, Proposition 5.6)) . Let P be a G -principal bundle and let H be a Liesubgroup of G . Then G has a reduction to a principal H -bundle if and only if the associated bundle P/H → M admits a section.In particular, if H = { e } then P has a section. Corollary 4.
Let ¯ P ( M, ¯ G, π ) a principal G -bundle with singularities, and let Σ be the set of singu-larities. Then there is a section s : M \ Σ → ¯ P /G . In what follows by ¯ g we denote the Lie algebra of ¯ G , and by g the Lie algebra of G . Moreover, wesuppose that the Lie group ¯ G is compact and connected. Then G is also compact and the quotient¯ G/G is reductive, i. e, ¯ g has an ad ( G )-invariant decomposition ¯ g = g ⊕ m .Let M be a 2-dimensional compact connected oriented manifold. Let ¯ P ( M, ¯ G, ¯ π ) be a principal G -bundle with singularities, where the set of singularities Σ consists of isolated points. Consider thefiber bundle π E : E = ¯ P /G → M . Recall that the canonical projection π G : ¯ P → E = ¯ P /G is aprincipal G -bundle [8](Chapter I, Proposition 5.5), and π G is a fiber bundle morphism:(74) ¯ P π G (cid:47) (cid:47) π (cid:31) (cid:31) E = ¯ P /G π E (cid:122) (cid:122) M Assume that ¯ P is endowed with a connection form ¯ ω : T ¯ P → ¯ g , and let ¯ H = ker ¯ ω be the horizontaldistribution of the connection, and ¯ V = ker dπ the vertical distribution on ¯ P . Note that ¯ H and ¯ V are vector subbundles of T ¯ P , and T ¯ P = ¯ H ⊕ ¯ V .Define the operator field (cid:101) ω : T ¯ P → ¯ V as follows:(75) (cid:101) ω ¯ p ( ¯ X ) = σ ¯ p (¯ ω ¯ p ( ¯ X )) , where ¯ X ∈ T ¯ p ¯ P , and σ ¯ p (¯ a ), ¯ a ∈ ¯ g , is the value of the fundamental vector field σ (¯ a ) at the point ¯ p . Lemma 3. a) (cid:101) ω is a field of projectors onto the vertical distribution ¯ V with kernel ¯ H .b) (cid:101) ω ¯ p ¯ g ◦ dR ¯ g ( ¯ X ) = dR ¯ g ◦ (cid:101) ω ¯ p ( ¯ X ) , where R ¯ g : ¯ G → ¯ G , R ¯ g ¯ g (cid:48) = ¯ g (cid:48) ¯ g .Proof. It is clear that (cid:101) ω vanishes on ¯ H . For ¯ X ∈ ¯ V ¯ p , ¯ p ∈ ¯ P , one can find ¯ a ∈ ¯ g such that σ ¯ p (¯ a ) = ¯ X .Then, ¯ ω ¯ p ( ¯ X ) = ¯ a , hence follows that (cid:101) ω ¯ p ( ¯ X ) = σ ¯ p ( a ) = X . Thus we get a).Now a) implies b) because dR ¯ g : T ¯ p ¯ P → T ¯ p ¯ g ¯ P is an isomorphism, which maps ¯ H ¯ p to ¯ H ¯ p ¯ g , and ¯ V ¯ p to ¯ V ¯ p ¯ g . (cid:3) Statement 11. a) There exists a operator field ω E on E such that, for each ¯ p ∈ ¯ P , the followingdiagram is commutative: (76) T ¯ p ¯ P (cid:101) ω (cid:47) (cid:47) dπ G (cid:15) (cid:15) ¯ V ¯ pdπ G (cid:15) (cid:15) T π G ( p ) E ω E (cid:47) (cid:47) V E π G ( p ) b) ω E is a projector onto V E with kernel HE = ker ω E = dπ G ( ¯ H ) .c) The decomposition T ¯ P = ¯ H ⊕ ¯ V of T ¯ P projects onto a decomposition T E = HE ⊕ V E of T E via dπ G . GENERALIZATION OF THE GAUSS-BONNET AND HOPF-POINCAR´E THEOREMS 19
Proof. a) If dπ G ( ¯ X ¯ p ) = dπ G ( ¯ X (cid:48) ¯ p (cid:48) ), then ¯ p (cid:48) = ¯ pg , and ¯ X (cid:48) = dR g ¯ X + ¯ Y , where g ∈ G and ¯ Y = σ ¯ pg ( a ) ∈ ker dπ G with a ∈ g . By Lemma 3, we have(77) (cid:101) ω ¯ pg ( ¯ X (cid:48) ) = (cid:101) ω ¯ pg ( dR g ¯ X + σ ¯ pg ( a )) = (cid:101) ω ¯ pg ( dR g ¯ X ) + σ ¯ pg ( a ) = dR g ( (cid:101) ω ¯ p ( X ) + σ ¯ pg ( a ) . Then,(78) dπ G ( (cid:101) ω ¯ p ( ¯ X )) = dπ G ( (cid:101) ω ¯ p (cid:48) ( ¯ X (cid:48) )) , this proves that ω E in the diagram (76) is well defined.b) From the diagram (76) it follows that dπ G ( ¯ H ) ⊂ HE . For any X ∈ HE π G (¯ p ) , take ¯ X ∈ T π G (¯ p ) ¯ P such that dπ G ( ¯ X ) = X . Then dπ G ( (cid:101) ω ( ¯ X )) = 0, this means that (cid:101) ω ( ¯ X ) = σ ¯ p ( a ), where a ∈ g . Then (cid:101) ω ( ¯ X ) − σ ¯ p ( a )) = 0, therefore ¯ X ) − σ ¯ p ( a ) ∈ ¯ H ¯ p and dπ G ( ¯ X ) − σ ¯ p ( a ) ∈ ¯ H ¯ p ) = dπ G ( ¯ X ) = X . Thus HE ⊂ dπ G ( ¯ H ) and therefore HE = dπ G ( ¯ H ).For any X ∈ V E π G (¯ p ) , X = dπ G ( ¯ X ), where ¯ X ∈ ¯ V . Then, by Lemma 3 and (76), we have that ω E ( X ) = dπ G ( (cid:101) ω ( ¯ X )) = dπ G ( ¯ X ) = X .c) follows from b). (cid:3) Remark . The bundle E → M is the bundle associated with the principal ¯ G -bundle ¯ P → M withthe fiber ¯ G/G with respect to the left action of ¯ G on ¯ G/G . The distribution HE is the connectioninduced on the associated bundle by the connection ¯ H on ¯ P .As ¯ G is compact, the homogeneous space ¯ G/G is reductive, this means that we have ad ( G )-invariantdecomposition ¯ g = g ⊕ m . Lemma 4.
The vertical subbundle
V E is isomorphic to the bundle associated with the principal G -bundle π G : ¯ P → E with the fiber m with respect to the left action of G on m : L g ( a ) = ad ( g − ) a .Proof. The isomorphism V E ∼ = ¯ P × ¯ G ¯ G × ¯ g / g ∼ = ¯ P × G m follows from the fact that V E = ¯ P × ¯ G T ( ¯ G/G ) [9](Chapter IV, section 18.18) and that T ( ¯ G/G ) = ¯ G × ¯ g / g [10](Chapter 4, section 5). Inthe exact form the isomorphism is written as follows:(79) ¯ P × G m → V E, [¯ p, a ] (cid:55)→ dπ G ( σ ¯ p ( a )) . (cid:3) The Hopf-Gauss-Bonnet theorem for principal G -bundle with singularities. Let ¯ P → M be a principal G ⊂ ¯ G -bundle with singularities, where ¯ G is a compact Lie group. Let us takea connection ¯ H in ¯ P with the corresponding connection form ω . Let E = ¯ P /G and H be theconnection in E induced by ¯ H with the connection form ω E (see Statement 11 and Remark 9).Given a 1-cohomology class a ∈ H ( ¯ G/G ), by Statement 1 we can construct a 1-form α ∈ Ω ( E )on E such that(1) α | H = 0;(2) for each x ∈ M , di ∗ x α = 0 and [ i ∗ x α ] = H ( η x ) a .In this case one can construct the form α explicitly. As ¯ G/G is a homogeneous space of a compactLie group, we can take an invariant form ξ ∈ Ω inv ( ¯ G/G ) ∼ = Λ ( m ) such that [ ξ ] = a (see, e. g., [11],Ch.1, 1). Note that ξ is determined by its value ξ : m → R at [ e ] ∈ ¯ G/G . Then from Lemma 4 itfollows that ξ defines a form (cid:101) α ∈ Ω ( V E ):(80) (cid:101) α ([¯ p, a ]) = ξ ( a ) . Theorem 4.
Let ω E : T E → V E be a connection form induced by a connection in the principal ¯ G -bundle ¯ P → M (see Statement 11).Then (81) α ( X ) = ˜ α ( ω E ( X )) The form α given by (81) satisfies the properties (1) and (2) of Statement 1.Proof. For X ∈ H , ω E ( X ) = 0, therefore α ( X ) = 0.Recall that the trivializations of E = P/G are constructed in the following way. For a goodcovering U i of M we take sections s i : U i → P , then we define the charts(82) ψ i : U i × ¯ G/G → π − E ( U i ) , ( x, [¯ g ]) → [ s i ( x )¯ g ] = π G ( s i ( x )¯ g ) . Therefore, for x ∈ U i , η − x : ¯ G/G → E x = π − E ( x ), η − x [¯ g ] = π G ( s i ( x )¯ g ).Now let us fix x ∈ U i ⊂ M and let ¯ p = s i ( x ). For any y ∈ E x and Y ∈ V y E , we have that y = π G (¯ p ¯ g ) = [¯ p, ¯ g ] = η − x (¯ g ) and Y = dπ G ( σ ¯ p ¯ g ( a )) = [¯ p ¯ g, a ] = dη − x ( dL ¯ g a ), where a ∈ m , and L ¯ g : ¯ G/G → ¯ G/G , L ¯ g ¯ g (cid:48) = ¯ g ¯ g (cid:48) . Note that here we use the isomorphism constructed in Lemma 4.Then(83) α ( Y ) = (cid:101) α ( ω E ( Y )) = (cid:101) α ( Y ) = ξ ( a ) = ξ ( dL ¯ g a ) . Therefore α ( dη − x ( dL ¯ g a )) = ξ ( dL ¯ g a ), thus ( η − x ) ∗ α | E x = ξ , then η ∗ x ξ = α | E x .Thus d (cid:0) α | E x (cid:1) = 0, and (cid:2) α | E x (cid:3) = H ( η x ) a . (cid:3) The curvature form Ω E = Dω E of the connection ω E (see Section 3, equation (25)) can be repre-sented by the following equality:(84) Ω E = 1 / ω E , ω E ] , where [ , ] : Ω k ( E, T E ) × Ω l ( E, T E ) → Ω k + l ( E, T E ) is the Frolicher-Nijenhuis bracket [9](ChapterIV, section 16.3). In particular, for 1-forms ω and ω this bracket is given by the following equation[ w , w ]( X, Y ) = [ ω ( X ) , ω ( Y )] − [ ω ( Y ) , ω ( X )] − ω ([ ω ( X ) , Y ]) ω ([ ω ( X ) , Y ]) − [ ω ( Y ) , X ] − [ ω ( Y ) , X ] + ( ω ◦ ω + ω ◦ ω )[ X, Y ] . We repeat the arguments of subsection 3.4 and get that(85) dα = L X α + 1 / α ([ ω E , ω E ]) . Thus, by assuming the existence of the 2-dimensional oriented compact manifold S and the map q : S → E satisfying the properties given in Section 1, by Theorem 2, we obtain the followingequality(86) (cid:90) S ( q ∗ L X α + 12 q ∗ ˜ α ([ ω E , ω E ])) = (cid:88) i =1 ,k index x i ( s, a ) , where s : M \ Σ → E is the section corresponding to the reduction of ¯ P | M \ Σ to G . The resultgiven by the equation (86) will be called the Hopf-Gauss-Bonnet theorem for principal G -bundle withsingularities . GENERALIZATION OF THE GAUSS-BONNET AND HOPF-POINCAR´E THEOREMS 21
Remark . If X is a vector field on M , and Σ is the set of zeros of X , then Y = X | X | defines asection of the unit tangent bundle M → T M over M \ Σ. Since T M is the associated bundle tothe bundle of orthonormal frames M → SO ( M ) with respect of standard action of SO (2) on S ,then SO ( M ) | M \ Σ reduces to the trivial subgroup G = { e } of ¯ G = SO (2). Now, ¯ G/G = SO (2), and SO (2) is diffeomorphic to S , then H ( ¯ G/G ) is 1-dimensional vector space generated by the angleform on S . Therefore, we can choose this generator to construct the form α ∈ Ω ( SO ( M )), and theconnection form on SO ( M ) induced by the Levi-Civita connection. This connection form is given byequation (42). Thus, if S and q : S → T M are the manifold and the map, respectively, constructedin section 4, the equation (86) reduces to the equation (46). Hence, Hopf-Gauss-Bonnet theorem inprincipal G -bundle with singularities is a generalization of the classical Gauss-Bonnet theorem.5.3. Hopf-Gauss-Bonnet theorem for the case of the sum of Whitney of vector bun-dles.
Let E , E , · · · , E k be k vector bundles of rank 2 over a compact connected Riemanniantwo-dimensional manifold M , and for each i , let s i : M → E i be the zero section of E i . Let˜ E i = E i \ s i ( M ).For all i let us take sections s i : M → E i . Then s i is a section of ˜ E i with singularities, whose set ofsingularities is the set Σ i of zeros of s i . Thus, if we consider the product bundle ˜ E = ˜ E × ˜ E ×· · ·× ˜ E k over M \ Σ, where Σ = ∪ Σ i , then the map s = ( s , · · · , s k ) : M \ Σ → ˜ E is a section with singularitiesof ˜ E . In addition, suppose that the section s : M \ Σ → ˜ E admits a resolution of singularities (seesection 1). This means that there exists a 2-dimensional oriented compact manifold S with boundary ∂S = (cid:83) j =1 ,k S j , and a map q : S → ˜ E such that q restricted to ◦ S = S \ ∂S is a diffeomorphismbetween S \ ∂S and s ( M \ Σ).Let us take connection forms ω i : T ˜ E i → V ˜ E i on ˜ E i , for i = 1 , · · · , k , then ω = ( ω , · · · , ω k ) is aconnection form on E . The curvature form of ω is Ω = (Ω , · · · , Ω k ), where Ω i is the curvature formof ω i .Since the fiber of the bundle ˜ E i is F i = R \{ } , and the cohomology class a i = [ dφ i ] represented bythe angular form on F i is a generator of H ( F i ), then the 1-form ( dφ , · · · , dφ k ) represents a nonzerocohomology class a ∈ H ( F ), where F = F × · · · × F k is the fiber of the bundle E . Note that F retracts by deformation to the k -torus T k . With this in mind, we can choose the 1-form α ∈ Ω ( E )satisfying properties (1) and (2) of Statement 1 as follows:(87) α = ω + · · · + ω k , where ω i = dϕ i + π ∗ G i is the connection form of ˜ E i (compare with (42)).Then dα = π ∗ dG , where G = (cid:80) ki =1 G i , and by applying Theorem 2 we obtain that(88) (cid:90) S q ∗ π ∗ dG = (cid:88) x i ∈ Σ index x i ( s, a ) . Let us fix an area form θ on M . Let dG i = K i θ , and call the function K i the curvature of theconnection ω i .If we consider the points of Σ \ Σ i as singular points of the section s i , then it is a section withsingularities of the bundle ˜ E i → M . In this sense we have the following statement. Theorem 5.
Let M be a compact connected two-dimensional manifold, let E i → M, i = 1 , · · · , k ,are k vector bundles of rank over M , let ˜ E i = E i \ s i ( M ) , where s i : M → E i is the zero sectionof E i , and let s i : M → E i be a section of E i for each i . If ω , · · · , ω k are connections on E , · · · , E k respectively, K , · · · , K k are the curvatures of these connections, and s = ( s , · · · , s k ) : M \ Σ → ˜ E ,then (89) (cid:88) j =1 ,k (cid:90) M K i θ = (cid:88) x i ∈ Σ (cid:88) j =1 ,k ind x i ( s j , [ dφ j ]) , where dφ i ∈ Ω ( R ) is the angular form of R , and θ is the area form on M .Proof. Let S be an oriented 2-manifold with boundary, and let q : S → E a map which satisfies thecondition given in section 1, that is, q i restricted to ◦ S = S \ ∂S is a diffeomorphism between S \ ∂S and s i ( M \ Σ). Now, because E = ˜ E × · · · ˜ E k , q has the form q = ( q , · · · , q k ), and for each i , themap q i : S → ˜ E i also satisfies this condition. This fact follows from the commutative diagram(90) M \ Σ id (cid:47) (cid:47) s (cid:15) (cid:15) M \ Σ s i (cid:15) (cid:15) ◦ S q (cid:47) (cid:47) s ( M \ Σ) ˜ p i (cid:47) (cid:47) s i ( M \ Σ)Since the vertical arrows are diffeomorphisms, we have that q i is a diffeomorphism. Therefore, foreach i = 1 , · · · , k , the map q i = ˜ p i ◦ q is also a diffeomorphism.Let us take the cohomology class a = [( dφ , · · · , dφ k )] of the fiber of ˜ E , where dφ i is the angularform on the fiber F i of E i , then a = [ dφ ] + · · · + [ dφ k ]. Thus, the expression at the right side ofequation (86) can be written as follows(91) (cid:88) x i ∈ Σ index x i ( s, a ) = (cid:88) x i ∈ Σ (cid:88) j =1 ,k ind x i ( s j , [ dφ j ]) . Furthermore, d i G = K i θ , and dα = (cid:80) j =1 ,k π ∗ ( K i θ ). Therefore, the expression at the left side ofequation (86) reduces to the following equality(92) (cid:90) S q ∗ π ∗ dG = (cid:88) j =1 ,k (cid:90) M K i θ Hence, we obtain the following equation(93) (cid:88) i =1 ,k (cid:90) M K i θ = (cid:88) x i ∈ Σ (cid:88) j =1 ,k ind x i ( s j , [ dφ j ]) . (cid:3) Remark . Note that if Σ i ∩ Σ j is non empty, and x ∈ Σ i ∩ Σ j , then the indexes ind x ( s i ) and ind x ( s j )could be different, still in the case where the vector bundles E i and E j are the same. References [1] Raoul Bott and Loring W. Tu.
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Universidad de los Andes, Bogot´a, Colombia
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