higher transgressions of the pfaffian

HIGHER TRANSGRESSIONS OF THE PFAFFIAN

SERGIU MOROIANU

ABSTRACT. We define transgressions of arbitrary order, with respect to families of unit-vectorfields indexed by a polytope, for the Pfaffian of metric connections for semi-Riemannian metricson vector bundles. We apply this formula to compute the Euler characteristic of a Riemannianpolyhedral manifold in the spirit of Chern’s differential-geometric proof of the generalized Gauss-Bonnet formula on closed manifolds and on manifolds-with-boundary. As a consequence, wederive an identity for spherical and hyperbolic polyhedra linking the volumes of faces of evencodimension and the measures of outer angles.

1. INTRODUCTION

The classical Gauss-Bonnet theorem computes the Euler characteristic of a closed surface (M, g)by integrating on M the Gaussian curvature kg. When M has a smooth boundary, one must adda correction term involving the average of the geodesic curvature of the boundary ∂M → M .If M has corners, i.e., the boundary itself has isolated singular points, then the exterior angle ofthese corners must also be taken into account. This general formula reads

2πχ(M) =

∫M

kgνg −∫∂M

a · lg +∑p

∠out(p)

where a : ∂M → R is the geodesic curvature function with respect to the outer normal, lg isthe induced volume form on ∂M , and ∠out(p) is the outer angle at a corner p. This outstandingformula has been generalized to arbitrary dimensions by a sequence of authors including H.Hopf, H. Weyl, C. Allendoerfer, A. Weil, and S.S. Chern.

In the present paper we extend the Gauss-Bonnet formula of Allendoerfer and Weil [3] to generalcompact Riemannian polyhedral manifolds (theorems 6.1 and 6.5).

The outline of the paper is as follows. In the first two sections we review the Pffafian of thecurvature using the language of double forms. In Section 4 we define smooth polyhedral mani-folds and polyhedral complexes, and study their properties with respect to integration of forms.The category of polyhedral complexes allows one to bundle together the outer cones of facesof Riemannian polyhedral manifolds, the natural locally trivial bundles of spherical polytopeswhere the contributions of the faces are localized. Starting from Chern’s construction [6] of atransgression form, we introduce in Section 5 higher transgressions for the Pfaffian form on vec-tor bundles endowed with a nondegenerate bilinear form and a compatible connection. We show

Date: December 11, 2020.1

2 SERGIU MOROIANU

that the exterior differential of these transgressions can be computed as a sum of lower-ordertransgressions. We then apply our abstract transgression theorem in Section 6 to the case of Rie-mannian polyhedral manifolds. This formula has been proved with entirely different methods byAllendoerfer and Weil [3] for a particular class of polyhedral manifolds that we call regular. Forregular polyhedral manifolds, the Gauss-Bonnet formula in the even-dimensional case followsby iterating the transgression formula on the boundary strata. In the general case, we use a globalpolyhedral complex to transfer the transgressions onto the outer cones via the polyhedral Stokesformula. The odd-dimensional case is reduced to the even case by analyzing the Riemannianproduct with an interval.

In the final section, we particularize our formula to space forms:

Theorem 1.1. Let M be a d-dimensional compact polyhedral manifold of constant sectionalcurvature k, with totally geodesic faces. Then

(1)χ(M)

2=∑j≥0

∑Y ∈F(d−2j)

kjvol2j(Y )

vol(S2j)

∠outY

vol(Sd−2j−1)

where F (d−2j) is the set of faces of M of dimension 2j, Sk is the standard unit sphere in Rk+1,and ∠outY is the measure of the outer solid angle at the face Y .

By convention, for 2j = d, both the volume of S−1 and the exterior angle of the interior faceof M are defined to be 1. We deduce from this theorem identities for hyperbolic polyhedrainvolving the volumes of even-dimensional faces and their outer angles, including an extensionto the noncompact case where some - or all - vertices are ideal.

Historical note. For submanifolds in Rn, the Gauss-Bonnet formula was stated and proved byHopf [12] for hypersurfaces, and by Allendoerfer [2] for submanifolds of arbitrary codimension.Allendoerfer and Weil [3] derived their formula on Riemannian polyhedra mainly as a tool fordeducing the Gauss-Bonnet on closed Riemannian manifolds without assuming the existence ofan isometric embedding in Euclidean space (which we now know to exist by Nash’s embeddingtheorem [15], but was unknown at that time). Their proof is indirect, based on a series of results:a triangulation theorem for polyhedral manifolds, an additivity result for the geometric side ofthe formula, a proof for simplices embedded in some Euclidean space using Weyl’s tube formula,an embedding theorem for analytic simplices, predating Nash, due to Cartan, and the Whitneyanalytic approximation result.

In a series of papers [6], [7], Chern gave an entirely different proof of the Gauss-Bonnet formula,based on his transgression form for the Pfaffian lifted to the sphere bundle. His constructionyields simultaneously the necessary correction term on manifolds with boundary, i.e, manifoldswith corners of codimension 1.

We extend here Chern’s method to transgressions of higher orders (transgressions of transgres-sions), and use these transgressions to prove the Allendoerfer-Weil formula for a more generalclass of polyhedral manifolds than that of [3]. In particular, the additivity of the geometric term

HIGHER TRANSGRESSIONS OF THE PFAFFIAN 3

in the generalized Gauss-Bonnet becomes a corollary of our proof. We close in this way a circleof ideas going back to Gauss and Bonnet, Hopf and Weyl, Allendoerfer and Weil and, last butnot least, S.S. Chern. This paper is a tribute to those great mathematicians from the past.

2. TENSOR CALCULUS AND THE PFAFFIAN

We fix in this section the notation concerning vector bundles with metric connections, and wedevelop a formalism for multiplying vector-valued forms. Such a formalism was already usedby Walter [20] in his generalized Allendoerfer-Weil formula for locally convex subsets in a Rie-mannian manifold, and also by Albin [1].

2.1. The Pfaffian. Let (V, h) be a 2n-dimensional real vector space endowed with a nondegen-erate symmetric bilinear pairing of signature (k, 2n−k). This means that we can find orthogonalbases e1, . . . , e2n with

h(e1, e1) = . . . = h(ek, ek) = 1, h(ek1 , ek+1) = . . . = h(e2n, e2n) = −1.

If moreover V is oriented, the volume form defined by h is the unique 2n-form volh ∈ Λ2nV ∗

which takes the value 1 on any such basis. This form defines an isomorphism

R→ Λ2nV, 1 7→ volh.

The inverse Bh : Λ2nV → R of this isomorphism is called the Berezin integral with respect to h.

Any skew-symmetric endomorphism A ∈ End−(V ) determines a 2-form ωA ∈ Λ2V ∗ by

ωA(u, v) = h(u,Av).

The nth power of the 2-form ωA is a multiple of volh. Define the Pfaffian of A with respect to hby

Pf(A) = 1n!Bh [(ωA)n] .

Example 2.1. Let V = R2n with its euclidean metric and let e1, . . . , e2n be the standard basis.The Pfaffian is a polynomial with integer coefficients (in fact, ±1) in the n(2n− 1) independententries of A. It is well-known, and easy to prove, that Pf(A)2 = det(A).

(1) For n = 1, every skew-symmetric matrix takes the form A =

[0 a−a 0

]. Then

ωA = ae1 ∧ e2, Pf(A) = a.

(2) For n = 2 and A =

0 a b c−a 0 d f−b −d 0 g−c −f −g 0

,

ωA = ae1 ∧ e2 + be1 ∧ e3 + ce1 ∧ e4 + de2 ∧ e3 + fe2 ∧ e4 + ge3 ∧ e4,

Pf(A) = ag − bf + cd.

4 SERGIU MOROIANU

2.2. Vector-valued forms. Let M be a smooth manifold, and E1, E2, E3 real vector bundlesover M . To every linear map p : E1 ⊗ E2 → E3 we associate a “product” on spaces of vector-valued forms, i.e., a bilinear map

P : Ω∗(M,E1)× Ω∗(M,E2)→ Ω∗(M,E3),

(α1 ⊗ s1)× (α2 ⊗ s2) 7→ α1 ∧ α2 ⊗ p(s1, s2).

• A first example of such a product arises for E1 = R, E2 = E3 = E and p : R⊗ E → Ethe canonical isomorphism. We recover the Ω∗(M)-module structure of Ω∗(M,E).• When E1 = End(E), E2 = E3 = E, and p is the tautological pairing End(E)×E → E,

we recover the action of endomorphism-valued forms on E-valued forms.• A large class of examples arises when E1 = E2 = E3 = E are bundles of R-algebras,

and p is the algebra product of E.• As a particular case of the previous example, let E be the bundle of exterior algebras of

a vector bundle V . Set

Ωu,v(M,V ) := Ωu(M,ΛvV ∗).

When M and V are clear from the context we will suppress them, using simply thenotation Ωu,v. We get a bi-graded algebra structure on the space of double formsΩ∗(M,Λ∗V ) = ⊕u,vΩu,v.• Another particular case: take E to be the endomorphism bundle of some vector bun-

dle V . We get a composition product on the space of endomorphism valued formsΩ∗(M,End(V )).

The two last two examples may lead to confusion when V is additionally endowed with a non-degenerate symmetric bilinear pairing h. In that case, there is an identification of the space ofh-antisymmetric endomorphisms End−(V ) with Λ2(V ∗), given by

End−(V ) 3 A 7→ ωA ∈ Λ2(V ∗), ωA(u, v) = h(u,Av).(2)

So there exist two different “product” maps on Ω∗(M,Λ2(V ∗)): one is the “exterior product”taking values in Ω∗(M,Λ4(V ∗)), the other one is the “composition product” obtained by iden-tifying Ω∗(M,Λ2(V ∗)) with Ω∗(M,End−(V )) and then using the product of endomorphisms,hence taking values in Ω∗(M,End(V )).

For simplicity, in the sequel we shall write αβ for the “product” P (α, β) ∈ Ω∗(M,E3).

3. THE CURVATURE TENSOR AS A DOUBLE FORM

For every vector bundle E → M with connection ∇ and for every k ≥ 0 we denote by d∇ :Ωk(M,E) → Ωk+1 the exterior differential twisted by ∇ on E-valued forms. Recall that on atensor product S = α⊗s, where α ∈ Ωk and s ∈ Γ(E) are a locally-defined k-form, respectivelya section in E, d∇ is given by

d∇(S) = dα⊗ s+ (−1)kα∇s.


The first-order differential operator d∇ is a derivation on the graded Ω∗(M)-module Ω∗(M,E),in the sense that for any β ∈ Ωs(M) and S ∈ Ω∗(M,E), the Leibniz rule holds:

d∇(βS) = (dβ)S + (−1)sβd∇S.

The composition of two successive operators d∇ is a 0-th order differential operator, identifiedwith an element R∇ ∈ Ω2(M,End(E)). In other words, for all S ∈ Ω∗(M,E) we have

d∇(d∇S) = R∇S

where the product is induced from the canonical pairing End(E) × E → E. The tensor R∇ ∈Ω2(M,End(E)) is called the curvature endomorphism of∇.

When E has a nondegenerate symmetric bilinear pairing h preserved by ∇, the curvature endo-morphism R∇ will be skew-symmetric. As in (2), this endomorphism determines via h a doubleform

ωR ∈ Ω2,2 = Ω2(M,Λ2E∗), ωR(X, Y )(e1, e2) = h(e1, R∇(X, Y )e2)

called the curvature form. When no confusion can arise regarding h, we will continue to denotethe form ωR by R∇.

3.1. The Pfaffian of the curvature. For any real vector bundle E → M of even rank 2n en-dowed with a nondegenerate pairing h and a compatible connection ∇, we are ready to definethe Pfaffian form Pfh(∇). When E is oriented, multiplication by the volume form of h definesfor every d ≥ 0 an isomorphism

Ωd(M) 3 α 7→ α⊗ volh ∈ Ωd,2n.

The inverse of this isomorphism is called the Berezin integral Bh with respect to h. The Pfaffianform of∇ is defined as

Pfh(R∇) := 1

n!Bh(ωnR) ∈ Ω2n(M) = Ω2n,0.

When h is implicit from the context, for simplicity we write Pf(∇) for Pfh(R∇). Clearly, the

Pfaffian form vanishes identically if the dimension of the base M is smaller than the rank of E.Even if E is not oriented, we can still define the Pfaffian density of R, by using the above defi-nition locally with respect to any of the two possible orientations. This is well-defined becausevolh ⊗ volh is invariant under change of orientation.

Lemma 3.1. If there exists a ∇-parallel section s 6= 0 in E, then Pfh(R∇) vanishes identically

on M .

Proof. If∇s = 0, then R∇s = 0, hence ωnRs = 0, so Bh(ωnR) = 0.

Theorem 3.2. The Pfaffian of∇ is a closed form on M .

Proof. From the second Bianchi identity, d∇ωR = 0, so by the Leibniz rule d∇(ωnR) = 0. More-over, if α ∈ Ω∗,2n, then one can easily check that dBh(α) = Bh(d∇α). Hence

dPfh(∇) = 1n!dBh(ωnR) = 1

n!Bh(d∇ωnR) = 0.

6 SERGIU MOROIANU

With these preliminaries, we can now state the classical Gauss-Bonnet formula: Let (M, g) be acompact Riemannian manifold without boundary of dimension 2n, and R ∈ Ω2,2 the curvatureof a metric connection on TM . Then

(3)∫M

Pf(R) = (2π)nχ(M).

The formula is valid even if M is not orientable, in that case Pf(R) being a density. The readerinterested in a short proof of this statement can skip the next section dealing with polyhedralmanifolds and complexes thereof, as well as most of Section 5.

Remark. The Pfaffian form on the tangent bundle was introduced by H. Hopf [12] motivated bygeometric considerations that can be briefly summarized as follows: The infinitesimal volumeof the Gauss map of a compact hypersurface M ⊂ R2n+1 equals the determinant of the secondfundamental form, while the curvature of the hypersurface is, by the Gauss equation, the squareof the second fundamental form in the sense of double forms. It follows that the pull-back on Mthrough the Gauss map of the standard volume form from the sphere S2n equals the Pfaffian ofthe curvature. Hopf computed in this way the degree of the Gauss map intrinsically in terms ofthe integral of the Pfaffian on M .

4. POLYHEDRAL MANIFOLDS

A somewhat informal notion of polyhedral manifold was used in [3]. We give here a rigorousdefinition together with an extension to a larger category, that of polyhedral complexes.

Polyhedral manifolds extend the notion of manifolds with corners [14]. Since for manifolds withembedded corners the results of the current section become largely obvious, the reader interestedonly in such manifolds can proceed directly to Section 5.

4.1. Linear polyhedra.

Definition. Let V be a real vector space of dimension n, and SV ∗ = (V ∗ \ 0)/R∗+ the dualsphere, consisting of non-zero linear forms on V defined up to a positive constant. Let A ⊂ SV ∗

be a finite set. The open linear polyhedron in V defined by A is the cone

PA = x ∈ V ;α(x) < 0, (∀)[α] ∈ A.

A non-empty linear polyhedron is always unbounded according to this definition, since it isinvariant by dilations with positive factors. The condition α(x) < 0 is the same for every repre-sentative α ∈ [α], i.e., it is invariant under rescaling of α by a positive constant.

A closed linear polyhedron is the closure in V of an open linear polyhedron. The closure of PAis clearly

PA = x ∈ V ;α(x) ≤ 0, (∀)α ∈ Aand note that according to this definition, the interior in V of a closed polyhedron is nonempty.


Some of the linear forms defining a linear polyhedron PA may be redundant, so it is natural toconsider minimal sets of such defining forms.

Lemma 4.1. If a set A ⊂ SV ∗ defines a non-empty open linear polyhedron in V , there exists aunique minimal set A′ ⊂ A defining the same linear polyhedron, i.e., PA′ = PA. Two minimalsets A,A′ ⊂ SV ∗ define the same polyhedra in V if and only if they are equal.

Let PA be a nonempty open linear polyhedron defined by a minimal set of linear forms A. Forevery α ∈ A, the hyperface Pα

A is the open linear polyhedron inside the vector space ker(α) ⊂ Vdefined by the relations β| ker(α); β ∈ A, β 6= α. By minimality, this linear polyhedron is non-empty, has dimension n−1, and sits inside the closure of PA. The linear polyhedron PA has thusas many hypersurfaces as the cardinality of A. Note that the set of defining forms for Pα

A indexedby A \ α may be non-minimal.

Inductively, one defines the faces of depth (or codimension) l ≥ 2 of a linear polyhedron P asthe hyperfaces of the faces of depth l − 1 of P . A closed linear polyhedron is thus decomposedinto the disjoint union of its open faces. We denote

F(P ) =⋃l≥1

F (l)

the set of all faces of P of codimension at least 1.

Recall the Minkowski-Weyl theorem [21, 5]: every open linear polyhedron P can be describedalternately as the set of linear combinations with positive coefficients of some generating vectorsv1, . . . vk:

P = c1v1 + . . .+ ckvk; cj > 0, (∀)j = 1, . . . , k.A minimal set of such generating vectors is unique up to rescaling by positive constants. Con-versely, every finite set of vectors in V spans a linear polyhedron by positive linear combinationsas above. (This linear polyhedron could be the whole of V , corresponding to the empty set oflinear forms A = ∅).

If P = PA is an open linear polyhedron in V defined by a finite set of forms A ⊂ SV ∗, the dualpolyhedron P ∗ is the linear polyhedron inside the dual space V ∗ defined as the positive linearspan of the vectors α ∈ A. For every face F ⊂ P , the conormal spaceN∗F is the space of formsin V ∗ which vanish on F (or equivalently, on its linear span). The intersection of N∗F with P ∗

is a face of P ∗.

4.2. Polyhedra.

Definition 4.2. An open polyhedron in a vector space V is the set of points v ∈ V satisfying theinequalities

αj(v) < aj, j = 1, . . . , k(4)

for some α1, . . . , αk ∈ V ∗ and a1, . . . , ak ∈ R.

8 SERGIU MOROIANU

Closed polyhedra and their faces are defined as in the case of linear polyhedra. A polytope isa compact polyhedron. According to our definition, the interior in V of a closed polyhedron isalways nonempty, and its dimension is dim(V ).

Definition 4.3. The conormal outer cone F out of an open face F ⊂ P in a polyhedron P ⊂ Vis the set of forms α ∈ V ∗ such that for every v ∈ F and v0 ∈ P , α(v − v0) > 0.

If the polyhedron P is defined by (4) and F ⊂ P is the open face determined by

F = v ∈ V ;α1(v) = a1, . . . , αl(v) = al, αl+1(v) < al+1, . . . , αk(v) < ak

then F out can also be described as the open cone

F out = c1α1 + . . .+ clαl; cj > 0, (∀)j = 1, . . . , k

so in particular F out is a linear polyhedron in N∗F ⊂ V ∗. If we fix v ∈ F and v0 ∈ P , take

(5) F out1 = α ∈ F out;α(v − v0) = 1.

Then F out1 is an open polytope independent of v, and independent of v0 up to a canonical iso-

morphism. Moreover, R∗+F out1 = F out, or in other words F out is a cone with base F out

1 .

A subcomplex C ⊂ P in a closed polyhedron P is the union of some closed faces of P .

Definition 4.4. Fix v0 in the interior of a polyhedron P . Let P out ⊂ V ×V ∗ be the disjoint union

P out =⋃

F∈F(P )

F × F out.

Define also P out1 as

P out1 =

⋃F∈F(P )

F × F out1 .

Lemma 4.5. If P is a linear polyhedron in V then the sets P out and P out1 (defined with respect

to some fixed v0 ∈ P ) are polyhedral subcomplexes in P × P ∗ ⊂ V × V ∗.

Proof. Clearly P out is a subcomplex of the product polyhedron P × P ∗. As for P out1 , it is the

intersection of P out with the affine hyperplane (v, α);α(v0) = −1.

4.3. Polyhedral manifolds and complexes. It is natural to define a topological polyhedral man-ifold X of dimension n as a separated topological space locally homeomorphic to some closedpolyhedron of dimension n, i.e., every point x ∈ X has a neighborhood homeomorphic to anopen set in a closed polyhedron (depending on x) of dimension n. These local homeomorphismsare called charts, and a collection of charts covering X is an atlas. Since every polyhedron islocally homeomorphic to a linear polyhedron by an affine transformation, we could have usedlinear polyhedra as model spaces in this definition.


Example 4.6. Fix an inner product on V . An open spherical polytope in V is the intersection ofan open linear polyhedron P ⊂ V with the unit sphere in V . The closure of a spherical polytopeis a polyhedral manifold diffeomorphic to the polytope

P1 = v ∈ P ;α0(v) = −1

for some fixed α0 in the dual polyhedron P ∗. The diffeomorphism is given by radial projection.

A topological polyhedral complex inside a polyhedral manifold M is a space C so that the pair(M,C) is locally homeomorphic to some subcomplex inside some closed polyhedron. Moreprecisely, for every x ∈ C there exists a polyhedron Px, a subcomplex Cx ⊂ Px, and a homeo-morphism, called chart of complexes, from a neighborhood D ⊂ P of x to an open set U ⊂ Px,such that U ∩ C is mapped homeomorphically onto U ∩ Cx.

Let P ⊂ V , P ′ ⊂ V ′ be open polyhedra and C ⊂ P ,C ′ ⊂ P ′ subcomplexes. A map f : D ⊂C → C ′ is called smooth if for every x ∈ C there exists an open neighborhood U of x in V anda smooth map F : U → V ′ extending f |U∩C : U ∩ C → C ′ such that F (U ∩ P ) ⊂ P ′.

Definition. A polyhedral manifold (respectively a polyhedral complex) is is a topological poly-hedral manifold (respectively a topological polyhedral complex) endowed with a smooth atlas.

A point of depth l ≥ 0 in a polyhedral manifoldM is a point mapped to a point of depth l througha chart (hence through any chart). A open face of codimension l of M is a connected componentof the set of points of depth l. Such a face is clearly a smooth manifold of dimension n − l.However, its closure is in general not a polyhedral manifold of the same dimension!

For a face Y inside a polyhedron M , we denote by F (l)(Y ) the set of open faces of M whichhave codimension l ≥ 0 in Y .

Definition 4.7. A polyhedral manifold M is regular if the closure of every open face of M isagain a polyhedral manifold.

Linear polyhedra and spherical polytopes are examples of regular polyhedral manifolds; the re-gion in the plane bounded by a smooth segment self-intersecting orthogonally in its end-pointsis an example of non-regular polyhedral surface. Regular polyhedral manifolds are an exten-sion of the notion of manifolds-with-corners with embedded faces. The polyhedral manifoldsconsidered in [3] seem to be regular, although the authors are imprecise on this aspect.

4.4. Outer spheres and the outer cone complex. Let x be a point in a face Y of a polyhedralmanifold M . An interior vector at x is the tangent vector in 0 to a smooth curve c : [0, ε)→ Mwith c(0) = x.

A Riemannian metric on a polyhedral manifold M is a metric which, in any chart modeled by apolyhedron in a vector space V , extends to a smooth metric on an open set of V . Riemannianmetrics can be constructed on any polyhedral manifold using partitions of unity.

10 SERGIU MOROIANU

Definition 4.8. Let x ∈ M be a point in the Riemannian polyhedral manifold M , and let Y bethe unique open face of M of which x is an interior point. The outer cone at x, denoted Cout

x Y ,is the set of those vectors in TxM whose inner products with every interior tangent vector at xare non-positive. The outer sphere Sout

x Y is the set of unit vectors in the outer cone at x.

Every vector V ∈ Soutx Y is orthogonal to Y , so Sout

x Y is a subset of the normal sphere to Y at x(otherwise, the projection of V on Y would have positive inner product with V ). Thus Sout

x Y isa spherical polytope inside NxY , the orthogonal complement in TXM to TxY .

Proposition 4.9. Let ((M, gM) be a Riemannian polyhedral manifold of dimension n, Y ⊂ Man open face of codimension l, and a ∈ Y a fixed point. Then

SoutY =⊔x∈Y

Soutx Y ⊂ TM

is a flat, locally trivial bundle over Y with fiber type the polytope Y ∗ = Souta Y . If M is regular,

then SoutY is globally trivial, SoutY ' Y × Y ∗ .

Proof. Let a ∈M . There exists a connected domainM ⊃ D 3 a, a chart φ : D → P ⊂ Rn withvalues in a closed polyhedron P , and a Riemannian metric gP on Rn whose pull-back throughφ is gM . Let Y be the unique open face of M containing a, and denote by l its dimension.Then D ∩ Y is mapped into a face F ⊂ P of the same codimension. Clearly φ∗ : TM → TPis a bundle isomorphism over its image, and thus for every x ∈ D its restriction to the outerspheres φx : Sout

x Y → Soutφ(x)F is a linear isomorphism of spherical polytopes. To show that

SoutY is locally diffeomorphic to a product, it is thus enough to show that SoutF is canonicallyisomorphic to the product bundle over F with fiber type Sout

b F .

The image of CoutF through the isomorphism TP → T ∗P induced by gP is just F × F out.We have seen in Example 4.6 that after fixing an interior point in P , the bundle of spheres inF × F out becomes canonically diffeomorphic to the product polytope F × F out

1 .

Assume now thatM is regular. The closure of every open face is then embedded inM , and hencethe unit outer normal vector field to every open hyperface extends continuously to the closure. Itfollows that every closed face Y is a connected component of an intersection of hyperfaces. Forall x ∈ Y , the generating vectors of the spherical polytope Sout

x Y are the unit normals to distincthyperfaces of M , and they trivialize SoutY as in Example 4.6.

The contribution of a face Y ⊂ M in the Gauss-Bonnet formula will turn out to be given by anintegral on the total space of the bundle of outer spheres SoutY , or equivalently the integral on Yof a transgression of the Pfaffian of order codim(Y ).

Proposition 4.10. Let (M, gM) be a Riemannian polyhedral manifold of dimension n. Then theset of outer-pointing unit tangent vectors, denoted Mout, is a polyhedral complex of dimensionn − 1 inside TM . The open top-dimensional faces of Mout are the interiors of the outer spherebundles SoutY , where Y spans all the open faces of M of codimension l ≥ 1.


Proof. Suppose first that M is the closure of an open polyhedron in Rn defined by a minimal setof linear forms α1, . . . , αp ∈ A. If Y ⊂ M is a face of codimension l, let AY ⊂ A be the setof those αj’s which vanish along Y . After relabeling, we can assume that AY = α1, . . . , αk.Proposition 4.9 tells us that CoutY is isomorphic to Y × Y ∗, where the polytope Y ∗ is the outersphere at some fixed point a ∈ Y . Moreover, Sout

y Y is spanned as a spherical polytope by theunit vectors V1(y), . . . , Vk(y) dual via gM to α1 . . . , αk. It follows that Sout

y Y is diffeomorphicto the conormal outer cone Y out under the difeomorphism TM → T ∗M induced by the metricgM . Hence CoutY , respectively SoutY are diffeomorphic to Y × Y out, respectively to Y × Y out

1 .Thus the claim follows from Lemma 4.5.

In general, M is only locally diffeomorphic to a polyhedron P , thus Mout is locally diffeomor-phic to a subcomplex in TP , which by definition means that it is a polyhedral complex.

4.5. Boundaries of polyhedral complexes and the Stokes formula. It is straightforward todefine smooth differential forms, the exterior derivative, and restriction of forms to faces ofpolyhedral manifolds. We prove below that Stokes formula also continues to hold on polyhedralcomplexes, once we properly define the boundary of a face.

Let M be a polyhedral complex of dimension k ≥ 1 and assume that we fix an orientation onall the faces of M of dimension k and k − 1. Let F ⊂ M be an open face of dimension k − 1.We define an integer µM(F ), the multiplicity of F in M , counting how many times F appears asthe oriented boundary of faces of M , as follows: take x ∈ F and a connected chart in M nearx, φ : D → P , mapping a neighborhood of x into the k-skeleton of a polyhedron P . The chartinduces orientations on the k-dimensional faces Y1, . . . , Ys of P ∩ φ(D) whose closure containsthe image of x, and also on the unique face F ′ ⊂ P ∩ φ(D) of dimension k− 1 containing φ(x).For j = 1, . . . , s let νj be a vector field along F ′ pointing inside Yj . Let µx(F ′, Yj) ∈ ±1 be1 if the orientations on F ′ and Yj are compatible (i.e., νj, e1, . . . , ek+1) is a negatively orientedframe in TxYj whenever (e1, . . . , ek+1) is a positively oriented frame in TxF ′) and −1 otherwise.Define

µx(F,M) =s∑j=1

µ(F ′, Yj) ∈ Z.

This quantity is locally constant on the connected face F , hence it is a constant, denoted byµ(F,M) ∈ Z. We define the (k − 1)-boundary of M as the formal sum

∂k−1(M) =∑

F∈Skk−1M

µ(F,M) · F.

Note that for a polyhedral manifold M of dimension n with orientable interior, Mout is a poly-hedral complex with a natural orientation on the n− 1-dimensional faces, and

∂n−2Mout = 0.

12 SERGIU MOROIANU

Lemma 4.11. Let M be an oriented polyhedral complex of dimension k and ω ∈ Ω∗(M) acompactly supported form. Then ∫

M

dω =

∫∂k−1M

ω.

Proof. By local considerations involving partitions of unity, it is enough to prove the claim whenthe form ω is supported in a chart domain. We may therefore assume that M is a subcomplexof dimension k in a polyhedron P , and moreover that ω is supported in a small ball which onlyintersects those faces passing through its center. On the intersection of this ball with each openk-dimensional face Y we compute the integral of dω using the usual Stokes formula, obtainingthe integral of ω on the k − 1-dimensional faces F bounding Y , with a sign depending on thecompatibility between the orientations on Y and F . When summing over all Y for a fixed F , weeventually get µ(F,M) times the integral of ω on F .

5. TRANSGRESSION FORMS

Let X be an l-dimensional polytope (see Section 4 for the definition). Let E → M be a realvector bundle of rank 2n endowed with a pseudo-metric and a compatible connection over asmooth manifold M of arbitrary dimension.

Let V be a family indexed byX of unit-length sections inE overM , i.e., a section V : X×M →π∗2E such that h(V, V ) = 1. In particular, for every x ∈ X , V (x, ·) is a section in E, so V canbe viewed as a map V ∈ C∞(X,Ω0,1(M,E)) (remember that E is identified with E∗ using h).The connexion ∇ in E acts of such section-valued maps, and ∇V ∈ C∞(X,Ω0,1(M,E)) is afamily of (1, 1)-forms indexed byX . Let dXV be the differential onX of the Ω0,1(M,E)-valuedfunction V . In local coordinates,

dXV :=l∑

j=1

dxj ⊗ ∂xjV ∈ Ω1(X,Ω0,1(M,E)),

(dXV )l = l!dx1 ∧ . . . ∧ dxl ⊗ ∂x1V ∧ . . . ∧ ∂xlV ∈ Ωl(X,Ω0,l(M,E)).

For integers l, k, n satisfying 0 ≤ l ≤ 2k + 1 ≤ 2n− 1, define a universal constant c(n, k, l) by

c(n, k, l) = 2kk!(n−1−k)!(2k+1−l)! ∈ Q.

Definition 5.1. For l ≥ 0, we define the (l+ 1)th transgression of the Pfaffian with respect to thefamily V : X → Ω(0,1)(M) by

T (l+1)V =

∑l≤2k+1≤2n−1

c(n,k,l)l!

∫X

B[V (dXV )l (∇V )2k+1−lRn−1−k

]∈ Ω2n−l−1(M).

Here (dXV )l ∈ Ωl(X)⊗ Ω0,l(M,E), R ∈ Ω2,2(M,E), and ∇V ∈ C∞(X,Ω1,1(M,E)), so B isapplied to a volume form on X tensored with a form of degree 2n− l − 1 on M .


For l = 0, the first transgression of the Pfaffian of the Levi-Civita connection on the spherebundle of a Riemannian manifold was introduced by Chern [6].

Functoriality. Like the Pfaffian, the transgression forms are functorial: Let E → M be a vectorbundle with metric and connection, and Φ : N → M a smooth map. We equip Φ∗E with itspull-back metric and connection. For every family V of unit sections in E indexed by X , we geta family Φ∗V of unit sections in Φ∗E. In this framework,

T (l+1)Φ∗V = Φ∗T (l+1)

V ∈ Ω2n−l−1(N).

When V can be inferred from the context, we will write T (l+1)Y instead of T (l+1)

VYfor the trans-

gression form with respect to the restriction of V along a subcomplex Y of X . By functoriality,this notation does not lead to confusion.

We are now ready to prove our main result about transgressions.

Theorem 5.2. Let E → M be a vector bundle endowed with a semi-riemannian metric h anda compatible connection ∇, and V a family of unit-length sections in E → M indexed by anoriented polytope X of dimension l. Then

dT (l+1)X =

−T (l)

∂X for l = dimX ≥ 1,

−Pf(∇) for X = ∗, i.e., l = dim(X) = 0,

where T (l)∂X ∈ Ω2n−l(M) is the sum of the transgression forms corresponding to the oriented

hyperfaces of X , i.e., the transgression corresponding to the boundary cycle of X .

Proof. Let π : SE → M be the locally trivial bundle of unit (pseudo-)spheres in E with respectto h. The tangent bundle to SE contains the vertical tangent bundle to the fibers. There is anatural horizontal complement to this vertical bundle, defined by using the connection ∇: thehorizontal lift of a path γ : R → M at a point v ∈ Sγ(0)M is the parallel transport of v along γ.Thus∇ induces a splitting of TSE as

TSE = T vertSE ⊕ π∗TM.

In the vector bundle π∗E → SE we have the tautological section s of h-length 1, defined bysv := v ∈ Eπ(v) = (π∗E)v. This connection preserves the pull-back metric h, still denoted bythe same symbol.

Lemma 5.3. Let∇1 = π∗∇ be the pull-back connection in π∗E → SM . Then

(6) ∇1s = IT vertSE

in the sense that∇1Us = U for every U ∈ T vert

v SE = Tv(SvM) ⊂ Ev.

Proof. Essentially by definition, the canonical section s is parallel in horizontal directions withrespect to ∇1, so ∇1s is a vertical double form. Also by definition, the pull-back connectionis trivial in vertical directions, so we compute ∇1s = ds on each vertical pseudo-sphere SvE,where s becomes a map from SvE to the fixed vector space Ev.

14 SERGIU MOROIANU

The idea of computing the Pfaffian of ∇1 (going back to Chern [6]) is to modify ∇1 on SE sothat s becomes parallel. For this, define a vertical (1, 2)-double form α ∈ Ω1(SE,Λ2π∗E) by

α = s · ∇1s, α(U) = s ∧ U ∈ Λ2π∗E

for every U ∈ T vertSE, and α(H) = 0 for H ∈ T horSE horizontal. Denote by A the skew-symmetric endomorphism-valued 1-form associated to α via h as in (2):

α(U)(V,W ) = 〈V,A(U)W 〉h, (∀)U ∈ TSE, (∀)V,W ∈ π∗E,(7)

so α = ωA using the notation from (2). (We identify E with E∗ via the musical isomorphism interms of h, thus Λ2π∗E ' Λ2π∗E∗). Then

∇1s = −As.

For t ∈ R set

∇t := ∇1 + (1− t)A.

Since A is skew-symmetric, ∇t clearly preserves h. We compute ∇ts = −tAs = tIT vertSE ,hence∇0s = 0 and thus, by Lemma 3.1, Pfh(∇0) = 0. We shall recover the Paffian of∇1 as theintegral from 0 to 1 of the t-derivative of Pf(∇t).

Consider the connection D on the pull-back bundle π∗E → R× SE defined by

(8) D := dt⊗ ∂t(·) +∇t = dt⊗ ∂t(·) +∇1 + (t− 1)s · ∇1s.

This connection also preserves the (pull-back of the) metric h.

Let V : X × M → SE be a family of unit sections in E (i.e., for every x ∈ X, p ∈ M ,V (x, p) ∈ Ep is a unit-length vector). Consider the smooth map

Φ : R×X ×M → R× SE, Φ(t, x, p) = (t, V (x, p))

and let Φ∗D be the pull-back connection in the bundle Φ∗π∗E = π∗3E (where π3 : R×X×M →M is the projection on the third factor).

By the naturality of curvature, we have

Φ∗Pfh(D) = Pf(Φ∗D) ∈ Ω2n(X × R×M).

Integrating this Pfaffian in the X × R variables, we set

(9) TX :=

∫[0,1]×X

Pf(Φ∗D) ∈ Ω2n−l−1(M).

We shall show below that TX equals the transgression T (l+1)X from Definition 5.1; for the time

being, we compute

dTX =

∫[0,1]×X

dMPf(Φ∗D) ∈ Ω2n−l(M).


The Pfaffian form is closed on R × X ×M , so by Stokes formula on the polyhedral manifold[0, 1]×X ×M ,

dTX =

∫[0,1]×X

−dt ∧ ∂tPf(Φ∗D)− dXPf(Φ∗D)

=

∫0×X

Pf(Φ∗D)−∫1×X

Pf(Φ∗D)−∫

[0,1]×∂XPf(Φ∗D)(10)

where ∂X is the oriented sum of hyperfaces of X . Now D|0×SE = ∇0. We have seen abovethat Pf(∇0) = 0 because there exists a non-zero parallel section for ∇0 on SE, so by naturalityof the Pfaffian, Pf(Φ∗D) = 0 on 0 ×X ×M .

Similarly, D|1×SE = (0, V )∗π∗∇ = π∗3∇, so Pf(Φ∗D) = π∗3Pf(∇) on 1 × X ×M . Thispull-back form does not contain any dxj (where x1, . . . , xl are the euclidean variables on X).Hence for l > 0 the integral on 1 ×X of the second term also vanishes, while for l = 0 (i.e.,when X is a point) it reduces to −Pf(∇).

By definition, the third term from (10) is the sum of the transgressions TF corresponding to theoriented hyperfaces F of X , which we denote T∂X .

In order to show that TX defined in (9) is the same as the transgression T (l+1)X from Definition

5.1, we must compute in more detail the Pfaffian of Φ∗D. From (8),

RD = π∗R∇ − (t− 1)π∗R∇s ∧ s+ 1−t22∇1s · ∇1s+ dt⊗ s · ∇1s.

The · products above are in the sense of double forms. Using the somewhat imprecise but sug-gestive notation∇ for the connection π∗3∇ over R×X ×M , we get for the curvature of Φ∗D:

Φ∗RD = R∇ − (t− 1)R∇V ∧ V + 1−t22

(dXV +∇V )2 + dt⊗ V · (dXV +∇V ).

We proceed to analyze the nth power of this double form inside the space of double formsΩ2n,2n(R×X ×M,π∗3E). Since double forms of even bi-order form a commutative algebra, wetreat this power as a homogeneous polynomial of degree n in these four terms of degree (2, 2).We are only interested in those monomials containing the volume form of R × X , and clearlythose terms must contain precisely once the form dt. Thus, the term dt ⊗ V · (dXV +∇V ) ap-pears precisely once, and so (t−1)R∇V ∧V does not contribute at all (since it contains V whichalready appeared in the former term, while for a monomial of top fiber degree 2n to be nonzero,the section V may occur at most once.) In conclusion of this discussion, those monomials from(Φ∗RD)n containing the volume form of R×X are contained in

ndt⊗ V · (dXV +∇V )[R∇ + 1−t2

2(dXV +∇V )2

]n−1

.

Using the binomial formula, we write the above as

ndt⊗ V ·n−1∑k=0

(n−1k

) (1−t2)k

2k (dXV +∇V )2k+1(R∇)n−k−1

.

16 SERGIU MOROIANU

Apply the binomial formula to the term (dXV +∇V )2k+1, and retain only the term of top degreel in the X variables, since we need to integrate over X . Hence TX from (9) is computed as

n−1∑k=d l+1

2 e

∫ 1

0

(1− t2)kdt · n(n−1k

)2−k(

2k+1l

)1n!

∫X

B[V(dXV

)l(∇V )2k+1−l (R∇)n−k−1

].

This gives precisely the transgression from Definition 5.1, since∫ 1

0

(1− t2)kdt = 2k(2k−2)...2(2k+1)(2k−1)...3·1 = (2k)!!

(2k+1)!!.

6. THE GAUSS-BONNET FORMULA ON POLYHEDRAL MANIFOLDS

6.1. The Allendoerfer-Weil formula in even dimensions. LetM be a compact C∞ polyhedralmanifold of dimension 2n endowed with a Riemannian metric g. Let Y ⊂ M be a face ofcodimension l ≥ 0, and NY ⊂ TM|Y the normal bundle of Y inside M with respect to g. Thesecond fundamental form of this inclusion is the bilinear map

A : TY × TY → NY , A(U,W ) = ∇MU W −∇Y

UW,

hence A is a NY -valued double form of degree (1, 1) on Y . We construct from A its dual, asmooth section A∗ on NY in the pull-back from Y of the bundle of double forms:

A∗ ∈ C∞(NY ,Λ1,1(Y )), A∗(V ) = g(V,A).(11)

For any B,C ∈ Ω1(Y, T ∗Y ⊗NY ) pure tensors of the form

B = b1 ⊗ b2 ⊗ ν1, C = c1 ⊗ c2 ⊗ ν2

define the partial contraction with g by

g(B,C) = g(ν1, ν2)(b1 ∧ c1)⊗ (b2 ∧ c2) ∈ Ω2,2(Y ).(12)

This definition allows us to define by linearity g(A,A) ∈ Ω2,2(Y ).

We are now ready to prove the extension of the Allendoerfer-Weil formula [3] for the Eulercharacteristic of a compact Riemannian polyhedral manifolds.

Theorem 6.1. Let M2n be a compact Riemannian polyhedral manifold. Then

(2π)nχ(M)−∫M

Pf(R) =2n∑l=1

n∑k=d l

2e

(−1)l2k−1(k−1)!(n−k)!(2k−l)!

∑Y ∈F(l)(M)

∫SoutY

BY[(RY − 1

2g(A,A))n−k(A∗)2k−l] |dg|.


Here |dg| is the family of spherical volume forms induced by g on the fibers of the normalsphere bundle SY → Y , while the Berezin integral produces a volume form on Y . The aboveintegral can thus be considered either as an integral on the total space of SoutY , or (using Fubini’stheorem) as an iterated integral, first along the fibers of SoutY → Y and then on Y . The symbol⌈l2

⌉denotes the smallest integer greater than or equal to l/2. We use the convention 0! = 1.

Also, the 0th power of a tensor like A∗ (for 2k = l) or RY − 12g(A,A) (for k = n) is understood

to be always 1, regardless of the vanishing of the tensor in question.

Proof. We first give the argument under the assumption that the polyhedral manifoldM is regular(Definition 4.7), thus recovering the main result of [3]. The general case requires some additionalcombinatorial properties of the outer cone complex and will be treated in Section 6.2.

We apply successively the transgression formula from Theorem 5.2 to the vector bundle TMrestricted to the various faces of M in increasing order of codimension.

An outer vector field with nondegenerate zeros. Starting from the lowest dimensional faces ofM ,we construct a smooth vector field along the boundary faces of M such that for every boundarypoint x inside the interior of a face Y , Ux lives in the outer sphere Sout

x Y . We extend this vectorfield smoothly to the interior of M , and perturb it to a vector field U ∈ Ω0(M,TM) transverseto the zero section M . If the perturbation is small enough in C0 norm, the vector field U will stillpoint in the outer sphere directions at every boundary face.

Define a unit vector field V0 := |U |−1U on the complement on the (isolated) zero-set Z(U) of Uin M . It is a section of the sphere bundle SM →M over the complement of Z(U).

Blow-up of the singular set of V0. Let M be the closure of V0(M \ Z(U)) in the polyhedralmanifold SM .

Remark. When M is a manifold with corners, the compact polyhedral manifold M is naturallydiffeomorphic to [M ;Z(U)], the total space of the blow-up of Z(U) inside M . See [14] for thenotion of real blow-up of manifolds-with-corners.

The boundary of the compactification M in SM is obtained by gluing the tangent sphere SpMnear each annulation point p ∈ Z(U). More precisely, besides the diffeomorphic image throughV0 of the boundary hyperfaces of M , ∂M contains also the “inner boundary”, i.e., the singulardivisor obtained by blowing-up the annulation points of U . Near a non-degenerate annulationpoint p ∈ Z(U), there exist local coordinates in which the vector field U takes the form

U(x) = x1∂x1 + . . .+ xr∂xr − xr+1∂xr+1 − . . . x2n∂x2n .

where the integer r is the index of U at p. The new hypersurface introduced by blowing up p isjust the compact manifold SpM , with orientation (−1)r+1 times the standard orientation induced

18 SERGIU MOROIANU

from TpM . We thus separate the boundary of M as

∂M =

⊔p∈Z(U)

SpM

t V0(∂M).

into the union of the inner boundary spheres, and the diffeomorphic image through V0 of theboundary of M .

In order to compute the integral of the Pfaffian on M , we will apply Theorem 5.2 to the pull-back bundle π∗TM → M over the compact polyhedral manifold M , endowed with the pull-backconnection π∗∇. This clever construction (due to Chern [6]) is necessary because M \ Z(U) isnot compact, so the Stokes formula would need to take into account the contribution of thesingularities Z(U) in the transgression forms. The role of the blow-up space M is precisely to”resolve” this singularity formally.

Since Z(U) is a finite set, it has measure 0. By naturality, the integral of the Pfaffian on M canbe computed by pull-back on M :

(13)∫M

Pf(R) =

∫V0(M\Z(U))

π∗Pf(R) =

∫M

Pf(Rπ∗∇).

Let T (1)(V0) ∈ Ω2n−1(M \ Z(U)) be the first-order transgression on M \ Z(U) from Defini-tion 5.1 corresponding to the unit vector field V0 interpreted as a 0-dimensional simplex of unitvector fields. Similarly, let T (1)(s) ∈ Ω2n−1(SM) be the first-order transgression from Defi-nition 5.1 corresponding to the canonical unit section s in π∗TM over M ⊂ SM , interpretedas a 0-dimensional simplex of unit vector fields. By naturality, π∗T (1)(V0) = T (1)(s) on thecomplement of the zero set Z(U). Now ∂M = V0(∂M)− tp∈Z(U)SpM . By Stokes formula,∫

M

Pf(Rπ∗∇) = −∫∂M

T (1)(s)

= −∫∂M

T (1)(V0) +∑

p∈Z(U)

∫SpM

T (1)(s).

Lemma 6.2 (Chern [6]). At a annulation point p ∈ Z(U) of index r, the integral on the sphereSpM of T (1)(s) equals (−1)r(2π)n.

Proof. Apply the definition of the transgression in dimension l = 0 for the canonical unit vectorfield s over M ⊂ SM . Here the parameter space X is just a point, hence the terms containingdXV vanish. The curvature R vanishes on the vertical sphere SpM since the connection ispulled-back from the base, hence the terms with k < n− 1 also vanish. It follows that

T (1)(s)|SpM = 2n−1(n−1)!(2n−1)!

B[s (π∗∇(s))2n−1]

where π∗∇(s) is given by (6).


It follows from this lemma and the Poincare-Hopf theorem that the inner boundary contributionsadd up, like in the boundary-less case, to (2π)nχ(M). We thus rewrite (13) as

(14) (2π)nχ(M) =

∫M

Pf(R) +

∫∂M

T (1)(V0).

This identity finishes the proof of the Gauss-Bonnet theorem for closed manifolds.

If M is a compact manifold with boundary, we can choose V0 to be the unit outer normal to ∂M .The correction term T (1)(ν∂M) is computed in that case as in the final part of the present proof.

When M has faces of codimension at least 2, in Eq. (13) the contribution∫YT (1)(V0) of a

boundary hyperface Y depends on the vector field V0, which cannot be chosen to be the unitnormal to Y simultaneously for all hyperfaces Y . In order to write this contribution in terms ofthe outer unit normal vector field νY , we use the higher transgressions with respect to the familiesof unit vector fields VY , V0,Y constructed in Section 4.4. By Theorem 5.2,

(15)∫Y

T (1)(V0) =

∫Y

T (1)(VY )−∫∂Y

T (2)(V0,Y ).

The point is, the first term in the right-hand side does not depend on V0, while the second is nowlocalized to codimension 2 faces. The induction procedure is powered by the next result:

Lemma 6.3. Let Y = F1 ∩ . . . ∩ Fl be a (possibly disconnected) face of M , where F1, . . . , Flare hyperfaces of M . Let Zj := ∩i 6=jFi. Then

l∑j=1

(−1)j∫Y

T (l)(V0,Zj) =

∫Y

T (l)(VY )−∫∂Y

T (l+1)(V0,Y ).

Proof. Direct application of Stokes formula and Theorem 5.2.

Again by induction, for all d ≥ 0 we have∫M

Pf(R) = (2π)nχ(M) +d∑l=1

∑Y ∈F(l)(M)

∫Y

T (l)(VY )

+∑

Y ∈F(d)(M)

∫∂Y

T (d+1)(V0,Y ).

The initial step is Eq. (15). Specializing to the maximal codimension d = 2n + 1, we havecompletely eliminated the non-canonical vector field V0 from the formula!∫

M

Pf(R) = (2π)nχ(M) +2n∑l=1

∑Y ∈F(l)(M)

∫Y

T (l)(VY ).

It remains to identify the contribution of each face in terms of intrinsic and extrinsic geometry ofthe faces (curvature and second fundamental form). Let Y be a face of codimension l ≥ 1. fromProposition 4.9, the map VY : Y ∗ × Y → SoutY is a trivialization of the outer sphere bundle of

20 SERGIU MOROIANU

Y , in particular it gives a family indexed by Y ∗ of unit vector fields in TM along Y . Since Y isfixed in this argument, we write V = VY for simplicity. The transgression T (l)(VY ) is defined by

T (l)(VY ) =n−1∑

k=d l2e−1

c(n,k,l−1)(l−1)!

∫Y ∗B[V (dY

∗V )l−1(∇V )2k+2−lRn−1−k] ∈ Ω2n−l(Y ).

For x ∈ Y , if νSoutY is the Riemannian volume form on the sphere Soutx Y , then (dY

∗V )l−1 can

be expressed as the pull-back on Y ∗ through the map V of the tensor square of the volume formof outer spheres:

(dY∗V )l−1 = (l − 1)!V ∗(νSout

x Y )⊗ νSoutx Y .

Thus the second component of the double form V (dY∗V )l−1 is a multiple of the volume form

of the normal bundle to Y . It follows that only those terms from ∇V and R whose secondcomponent is a form tangent to Y may have a nonzero contribution to T (l)(VY ). These terms areA∗(V ), the second fundamental form (11) of Y in M interpreted as a (1, 1) form-valued functionon NY , respectively R|Y , the components of the curvature form of M along Y . Recall that by theGauss equation

R|Y = RY − 12g(A,A)

where the contraction g(A,A) was defined in (12). For x ∈ Y we obtain by changing variablesin the integral from the polytope Y ∗ to the outer sphere Sout

x Y using the diffeomorphisms V :Y ∗ → Sout

x Y :

T (l)(VY )(x) =n−1∑

k=d l2e−1

c(n, k, l − 1)

∫Sout

x Y

νSoutx Y ⊗ 1·

B[1⊗ νNY

· (−A∗)2k+2−l(RY − g(A,A)/2)n−1−k] .

6.2. Passing from regular to general polyhedral manifolds. If M is a not regular polyhedralmanifold, the above proof breaks down because the outer cone bundles are not globally trivial.Thus we need a new global argument before applying the local computations from the previoussections. Let I denote the unit interval [0, 1].

In the pull-back of TM over the polyhedral manifold SM×Ix×It we consider the pull-back Dof the connection D from (8) under the projection off the factor [0, 1]x onto SM×It:

D = π∗∇+ (1− t)Awhere π is the projection SM×I2 → M , and A is the endomorphism-valued 1-form defined in(7) with respect to the tautological section s in π∗TM ,

A(W ) = s ∧ π∗W ∈ End−(π∗TM).

Here t is a deformation parameter as before, while x ∈ [0, 1] will be the variable of a conicaldeformation of the polyhedral complex Mout that we now introduce. Recall that we have fixed avector field U on M with isolated nondegenerate zeros and outward-pointing along ∂M , and weconstructed V0 = U/‖U‖ on the complement of Z(U). In particular, for every p ∈ Y ∈ F(M),


V0(p) belongs to the convex spherical polytope Soutp Y , where F(M) is the set of faces of M of

codimension at least 1.

For every face Y ∈ F(M) define a locally trivial bundle of spherical polyhedra

ConV0(SoutY ) = (cosx · vp + sinx · V0(p), x); p ∈ Y, vp ∈ SoutY, x ∈ I.

From Proposition 4.10, it follows that the set

ConV0(Mout) =

⋃Y ∈F(M)

ConV0(SoutY )

is a polyhedral complex embedded in M × I , so ConV0(Mout)× It is a polyhedral complex

embedded in M×I2. We enrich this complex by adding to it certain faces at x = 1. We startwith the image of V0, i.e., the face V0(M \ Z(U))×1×It of dimension 2n + 1. We completethis face with the 2n-dimensional cylinders SpM×1×It for each annulation point p ∈ Z(U).

Proposition 6.4. Let M be a polyhedral manifold of dimension 2n with orientable interior. Theset

P = ConV0(Mout)× I

⋃V0(M \ Z(U))×1×I

⋃p∈Z(U)

SpM×1×I ⊂ SM×I2

is a polyhedral complex of dimension 2n+ 1. Its 2n-boundary is

∂dim(M)P = V0(M)×1×1 − V0(M)×1×0

+∑

p∈Z(U)

SpM×1×I −∑

Y ∈F(M)

SoutY ×0 × I(16)

+∑

Y ∈F(M)

ConV0(SoutY )×1 −

∑Y ∈F(M)

ConV0(SoutY )×0.

In order to compute the integral on M of the Pfaffian of ∇, we are going to use the Pfaffian ofthe connection D. It is a closed form on SM × I2, hence by the Stokes formula on polyhedralcomplexes (Lemma 4.11),

∫∂P Pf(RD) = 0. Moreover, Pf(RD) vanishes identically on three of

the types of faces of ∂2nP from (16): It vanishes on ConV0(SoutY )×0 for Y ∈ F(M) and on

V0(M)×1×0 because at t = 0 the connection D admits a parallel section s. It also vanisheson ConV0(S

outY )×1 for Y ∈ F(M) because along t = 1 the connection D is the pull-backof ∇ from the base via the projection SM×I → M , hence by functoriality the Pfaffian Pf(RD)is a horizontal form. However, since dim(Y ) < 2n = rk(TM), the Pfaffian of∇ vanishes on Y .

In conclusion of this discussion, by integrating Pf(RD) on the polyhedral complexes from (16),we get after using (14) and pull-back by V0:∫

M

Pf(R∇) = (2π)nχ(M) +∑

Y ∈F(M)

∫SoutY×It

Pf(RD).

To conclude the proof, we note that the restriction ofD to x = 0 coincides with the connectionD defined in (8). Moreover, the computation of

∫SoutY×I Pf(RD), carried out above in the regular

case, is local in the base Y , so it remains valid even without the regularity assumption on M .

22 SERGIU MOROIANU

6.3. Odd dimensions. This case follows directly from the even-dimensional case as we nowexplain.

Theorem 6.5. Let (N, g) be a compact Riemannian polyhedral manifold of odd dimension 2n−1.Then

(2π)nχ(N) =2n−1∑l=1

n−1∑k=d l−1

2 e

(−1)l−1π(2k − 1)!!

(n− 1− k)!(2k + 1− l)!

·∑

Y ∈F(l)(N)

∫SoutY

BY[(RY − 1

2g(A,A))n−1−k(A∗)2k+1−l] |dg|.

By convention, (−1)!! = 0! = 0!! = 1, and the 0th power of a double form is always 1.

Proof. Apply theorem 6.1 to the product manifold M := N × I , where I is the interval [0, 1],endowed with the product metric h = g + dt2. The Euler characteristics of M and N coincide.We will exploit the fact that the vertical vector field ∂

∂tis parallel along M , but also along Y × I

for every face Y of N .

The boundary faces of M fall into two types:

• lateral faces of the form Y × I , and• top or bottom faces of the form Y × 0 or Y × 1.

The first type of faces do not contribute in the Gauss-Bonnet formula. Indeed, the curvature formRY×I and the second fundamental form AY×I of Y × I inside N × I both vanish in the directionof the parallel vector field ∂

∂t. It follows that the Berezin integral inside the term from theorem

6.1 corresponding to the face Y × I vanishes identically.

The outer spheres in M of the second type of faces, e.g. Y × 0, can be described as sphericalcones over the outer sphere of Y in N . More precisely, let V be an Euclidean vector space, Sthe unit sphere in V , V ′ a hyperplane in V and A ⊂ S ∩ V ′ a subset of S lying in a subsphere ofcodimension 1. Let p0, p1 = V ′⊥ ∩ S, so p0 and p1 are diametrally opposed and A sits in theequatorial hypersphere orthogonal to p0 and p1. We define the spherical cone of A with respectto p0 as the union of all geodesic segments in S linking p0 to A. The complement of the vertexp0, the open spherical cone, is isometric to a topological product A × [0, π/2) with the warpedproduct metric cos2(α)gA + dα2. Note for later use that the volume densities induced by g andgA on the fibers of the outer spheres satisfy the identity

(17) |dg| = cos(α)dim(V )−2dα|dgA|.

Lemma 6.6. Let Sout(Y ) be the outer sphere of Y in N , and Sout(Y × 0) the outer sphereof Y × 0 in M . Then for every x ∈ Y , Sout

x (Y × 0) is the spherical cone of Soutx (Y ) with

respect to the point ∂∂t

. Moreover, the union for all x ∈ Y of the open spherical cones form alocally trivial bundle with fiber type [0, π/2) over Sout(Y )


Using this lemma, we can carry out the integral in α (i.e., along the fibers of the spherical conefibration) of the integrands from theorem 6.1. For a fixed x ∈ Y , the curvatureRY and the metriccontraction of the second fundamental form g(A,A) are pull-backs from the base, i.e., they areconstant on the outer sphere. The dualA∗ of the second fundamental form is linear on the normalbundle to Y and vanishes at the vertical vector ∂t, hence for v ∈ Sout

x Y ,

A∗(cos(α)v + sin(α)∂t) = cos(α)A∗(v).

Now the volume form on the spherical cones is given by (17). It followsthat the push-forward along the fibers of the spherical cones over Sout

x Y ofBY[(RY − 1

2g(A,A))n−1−k(A∗)2k+1−l] |dgM | amounts to

I2kBY[(RY − 1

2g(A,A))n−1−k(A∗)2k+1−l] |dgN |

where I2k is a scaling factor independent of x:

I2k =

∫ π/2

−π/2cos2k(α)dα =

π(2k − 1)!!

(2k)!!.

We have used the convention 0! = 0!! = (−1)!! = 1.

7. CONSTANT-CURVATURE POLYHEDRAL MANIFOLDS WITH GEODESIC FACES

By applying the Gauss-Bonet theorems 6.1 and 6.5 to the case of polyhedral manifolds of con-stant sectional curvature with totally geodesic faces, we obtain certain identities for spherical,euclidean and hyperbolic polyhedra in terms of volumes of faces and measures of outer angles.

7.1. Euclidean polyhedra. Let M be a flat compact polyhedral manifold of dimension k withtotally geodesic faces. In this case, the Gauss-Bonnet simply states that the sum of the outerangles at the vertices of M equals the Euler characteristic χ(M) divided by the volume of thek − 1 sphere. Indeed, in Theorems 6.1 and 6.5 the curvature RY of the face Y and the secondfundamental form A of Y ⊂ M both vanish, so the only non-zero terms in the right-hand sidearises for codim(Y ) = k, i.e., when Y is a point. In that case, the integral corresponding to avertex Y gives the volume of the outer sphere at the vertex Y , and the formula becomes

vol(Sk−1)χ(M) =∑

V ∈F(k)

∠outY.

In particular, the Euler characteristic of a flat compact polyhedral manifold with totally geodesicfaces is always non-negative, and it is necessarily positive as soon as M has at least one vertex.This identity is clear for open polytopes in Rk, since the outer spheres of the vertices partitionthe unit sphere Sk−1 into spherical polytopes with mutually disjoint interiors. But in general it isnot obvious. A direct proof should rely on some additivity property of outer angles.

24 SERGIU MOROIANU

7.2. Manifolds of constant sectional curvature with geodesic faces. Let (M, g) be a compactpolyhedral manifold with constant scalar curvature k and with geodesic faces. Then R = k

2g2,

valid on every face. Since the second fundamental form A of any face Y is assumed to vanish,we also have A∗ = 0. Therefore Theorems 6.1 and 6.5 give

(2π)nχ(M) =n∑k=0

2k−1(k−1)!(n−k)!

∑Y ∈F2k(M)

∫SoutY

BY[(RY )n−k

]|dg| for dim(M) = 2n,

2nπn−1χ(N) =n∑k=1

(2k−3)!!(n−k)!

∑Y ∈F2k−1(N)

∫SoutY

BY[(RY )n−k

]|dg| for dim(N) = 2n− 1

where we recall that F (k) denotes the space of faces of codimension k ≥ 1 On a face Y ofdimension 2j, we compute moreover for R = RY

Rj = kj(2j)!

2jdgY ⊗ dgY , BY (Rj) = kj

(2j)!

2j|dgY |.

In conclusion, regardless of the parity of d = dim(M), the Gauss-Bonnet formula becomes thesum (1) over the even-dimensional faces of Md advertised in the introduction as Theorem 1.1.

Dehn [11] studied such identities for small dimensions and predicted their existence in general.Allendoerfer-Weil’s formula from [3] was used by Santalo [17] for deducing particular cases of(1) for polyhedra embedded in a constant curvature space-form. For a spherical simplex insideSk, the identity was also announced by Kenzi Sato [19].

7.3. Hyperbolic polyhedra with ideal vertices. As an extension of the previous example, thehyperbolic identity allows us to compute the volume of hyperbolic 2n-polyhedra with some – orall – ideal vertices in terms of outer angles and volumes of lower-dimensional faces, by passingto the limit the Gauss-Bonnet formula for compact polyhedra. For instance, when 2n = 4 thevolume of an ideal hyperbolic 4-simplex is given by

vol(M) = −2π2 +π

3

∑Y ∈F(2)(M)

∠out(Y )

where ∠out(Y ) is the outer dihedral angle of the ideal triangle Y in M , i.e., the angle betweenthe outer normals to the two hyperfaces containing Y .

8. CLOSING REMARKS

The Gauss-Bonnet formula (3) continues to hold on complete manifolds with warped productends with a decay condition on the warping functions [16], and for asymptotically cylindricalmetrics [1]. If (M, g) is a smooth compact Riemannian manifold-with-boundary, the Gauss-Bonnet formula contains a correction term along the boundary in terms of the second fundamen-tal form [3, 6]. Extensions of this formula to more general metrics on the interior of a manifold-with-boundary were found by Satake [18] for Riemannian orbifolds, by Albin [1], Dai-Wei [10]and by Cibotaru and the author [9] for manifolds with fibered boundaries, by McMullen [13] for


cone manifolds, by Anderson [4] for asymptotically hyperbolic Einstein 4-manifolds, and againin [9] for incomplete edge metrics, to cite only a few results in this direction. The proofs typicallystart from a degeneration process in the Gauss-Bonnet formula for manifolds-with-boundary.

In contrast, the Gauss-Bonnet formula on a Riemannian polyhedral manifold does not seem tofollow from such a degeneration. Although it may appear tempting to consider a ε-neighborhoodof M as a C1-smoothing of the boundary and then try to compute the limit of the boundaryintegrand as ε → 0 by interpreting the smoothed boundary as a current like in [8], we were notable to isolate with that approach the contributions of lower-dimensional faces.

REFERENCES

[1] P. Albin, A renormalized index theorem for some complete asymptotically regular metrics: The Gauss–Bonnettheorem, Adv. Math. 213 (2007), no. 1, 1–52.

[2] C. Allendoerfer, The Euler number of a Riemannian manifold, Amer. J. Math. 62 (1940), 243–248.[3] C. Allendoerfer, A. Weil, The Gauss-Bonnet Theorem for Riemannian Polyhedra, Trans. AMS, 53 (1943), no.

1, 101–129.[4] M. T. Anderson, L2 curvature and volume renormalization of AHE metrics on 4-manifolds, Math. Res. Lett. 8

(2001), no. 1–2, 171–188.[5] M. Beck, S. Robins, Computing the continuous discretely: integer-point enumeration in polyhedra, Springer

Undergrad. Texts Math, 2007.[6] S. S. Chern, A simple intrinsic proof of the Gauss-Bonnet formula for closed Riemannian manifolds, Ann.

Math. 45 (1944), 747–752.[7] S. S. Chern, On the curvature integra in a Riemannian manifold, Ann. Math. 46 (1945), no. 4, 674–684.[8] D. Cibotaru, Chern-Gauss-Bonnet and Lefschetz duality from a currential point of view, Adv. Math. 317

(2017), 718–757.[9] D. Cibotaru, S. Moroianu, Odd Pfaffian forms, preprint arXiv:1807.00239.

[10] X. Dai, G. Wei, Hitchin-Thorpe inequality for noncompact Einstein 4-manifolds, Adv. Math. 214 (2007), no.2, 551–570.

[11] M. Dehn, Die Eulersche Formel im Zusammenhang mit dem Inhalt in der Nicht-Euklidischen Geometrie, Math.Ann. 61 (1906), 561–586.

[12] H. Hopf, Uber die Curvatura integra geschlossener Hyperflachen, Math. Ann. 95 (1925), 340–376.[13] C. McMullen, The Gauss Bonnet theorem for cone manifolds and volumes of moduli spaces, Amer. J. Math.

139 (2017), 261–291.[14] R. B. Melrose, Analysis on manifolds with corners, book in preparation.[15] J. Nash, The Imbedding Problem for Riemannian Manifolds, Ann. Math 63, no. 1 (1956), 20–63.[16] S. Rosenberg, On the Gauss-Bonnet theorem for complete manifolds, Trans. AMS, 287, (1985) no. 2, 745–753.[17] L. Santalo, Sobre la formula de Gauss-Bonnet para poliedros en espacios de curvatura constante, Rev. Un.

Mat. Argentina 20 (1962), 79–91.[18] I. Satake, The Gauss-Bonnet Theorem for V-manifolds, J. Math. Soc. Japan 9 (1957), no. 4, 464–492.[19] K. Sato, Solid angles of high dimensional spheres, Kyoto U. Departmental bulletin paper (2006), 1503:129–

131.[20] R. Walter, A generalized Allendoerffer-Weil formula and an inequality of the Cohn-Vossen type, J. Differential

Geometry 10 (1975), 167–180.[21] H. Weyl, Elementare Theorie der konvexen Polyeder, Comment. Math. Helvetici 7 (1935), 290–306.

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