Flat extensions of principal connections and the Chern–Simons \(\boldsymbol{3}\)-form

1 Introduction

Chern–Simons forms and invariants derived from them are prominent examples of secondary invariants of certain types of connections [4]. Apart from their interest in geometry and topology, Chern–Simons forms also play a fundamental role in theoretical physics. In contrast to standard characteristic classes and characteristic numbers which are defined on even-dimensional manifolds, Chern–Simons invariants are defined for manifolds of odd dimension. Among Chern–Simons forms, much interest is devoted to the Chern–Simons \(3\)-form which is defined for a \(1\)-form \(\theta\) with values in the Lie algebra \(\mathfrak{g}\) of a Lie group \(G\) via \[\operatorname{CS}(\theta)=\langle\theta,\mathrm{d}\theta\rangle+\tfrac13 \langle\theta,[\theta,\theta]\rangle,\] where \(\langle\,\cdot\,,\cdot\,\rangle\) denotes a symmetric bilinear form on \(\mathfrak{g}\) which is invariant under the adjoint action \(\operatorname{Ad}\) of \(G\) and \([\,\cdot\,,\cdot\,]\) denotes the Lie bracket of \(\mathfrak{g}\). In the case where \(\theta\) is a principal connection on a trivial principal \(G\)-bundle \(\pi : P \to M\) over a closed oriented \(3\)-manifold \(M\), the Chern–Simons \(3\)-form can be used to assign a real number \[c_{\sigma}=\int_M \sigma^*\operatorname{CS}(\theta)\] to every global smooth section \(\sigma : M \to P\). Depending on the topology of \(G\), it may happen that \(c_\sigma\) is independent of \(\sigma\) or that one can choose \(\langle\,\cdot\,,\cdot\,\rangle\) so that \(c_{\sigma}\) is independent of \(\sigma\) up to addition of an integer. In the former case, one obtains a real valued Chern–Simons invariant associated to \(\theta\), in the latter – more frequent – case, an invariant with values in \(\mathbb{R}/\mathbb{Z}\).

The prototypical example of an \(\mathbb{R}/\mathbb{Z}\)-valued Chern–Simons invariant arises from considering the \(\mathrm{SO}(3)\)-bundle \(\pi : \mathcal{SO}M \to M\) of orientation compatible orthonormal frames of a closed oriented Riemannian \(3\)-manifold \((M,g)\), equipped with its Levi-Civita connection form \(\theta\) and where \(\langle\,\cdot\,,\cdot\,\rangle\) is a suitable scalar multiple of the Killing form of \(\mathfrak{so}(3)\). In [4], Chern–Simons show that the Riemannian \(3\)-manifold \((M,g)\) can be isometrically immersed into Euclidean \(4\)-space \(\mathbb{E}^4\) only if \(c_{\sigma}\) is an integer. Moreover, they observe that the associated \(\mathbb{R}/\mathbb{Z}\)-valued invariant is actually conformally invariant, so that the integrality of \(c_{\sigma}\) is an obstruction to the existence of a conformal immersion into \(\mathbb{E}^4\). The Chern–Simons invariant of a Riemannian \(3\)-manifold has since played an important role in hyperbolic geometry due to its relation to the \(\eta\)-invariant [11]. See also [9] and [8] for more recent related work.

In \(3\)-dimensional CR-geometry, the Chern–Simons \(3\)-form gives rise to an \(\mathbb{R}\)-valued Chern–Simons invariant as discovered by Burns–Epstein [1]. To include this and similar examples, we work in the setting of \(\mathfrak g\)-connections on principal bundles whose structure group is a subgroup of \(G\).

For differential geometric structures not related to Riemannian or CR-geometry, the meaning of the Chern–Simons invariant seems to have received less attention in the literature. In this article, we relate vanishing statements for the Chern–Simons invariant to the notion of a flat extension of a principal connection. To this end, suppose that \(G\) is a Lie subgroup of a Lie group \(\tilde{G}\) which is equipped with an \(\operatorname{Ad}\)-\(\tilde{G}\) invariant bilinear form \(\langle\,\cdot\,,\cdot\,\rangle\) on its Lie algebra \(\tilde{\mathfrak{g}}\). We assume that the restriction of \(\langle\,\cdot\,,\cdot\,\rangle\) to \(\mathfrak{g}\times \mathfrak{g}\) is non-degenerate so that \(\tilde{\mathfrak{g}}=\mathfrak{g}\oplus \mathfrak{g}^{\perp}\), where \(\mathfrak{g}^{\perp}\) denotes the orthogonal complement of \(\mathfrak{g}\) with respect to \(\langle\,\cdot\,,\cdot\,\rangle\). Writing a \(\tilde{\mathfrak{g}}\)-valued \(1\)-form \(\psi\) as \(\psi=\psi^{\top}+\psi^{\perp}\) with \(\psi^{\top}\) taking values in \(\mathfrak{g}\) and \(\psi^{\perp}\) taking values in \(\mathfrak{g}^{\perp}\), we define:

Let \(\operatorname{CS}(\mu_{\tilde G})\) denote the Chern–Simons form of the Maurer–Cartan form \(\mu_{\tilde G}\) of \(\tilde{G}\), computed with respect to \(\langle\,\cdot\,,\cdot\,\rangle\). Moreover, we compute the Chern–Simons \(3\)-form of a \(\mathfrak{g}\)-valued \(1\)-form with respect to the bilinear form on \(\mathfrak{g}\) obtained by restricting the bilinear form \(\langle\,\cdot\,,\cdot\,\rangle\) on \(\tilde{\mathfrak{g}}\) to \(\mathfrak{g}\times \mathfrak{g}\). In the case where the pair \((\tilde{\mathfrak{g}},\mathfrak{g})\) of Lie algebras is a symmetric pair (see Section 5 for details), we obtain:

Corollary 1.1 follows from of our slightly more general main Theorem 6.1. The proof of this theorem strongly depends on a certain algebraic identity – see Lemma 5.1 – which expresses \(\mathrm{CS}(\psi)\) as the sum of \(\mathrm{CS}(\psi^{\top})\) and a term involving the curvature of \(\psi\). In particular, if \(\psi\) satisfies the Maurer–Cartan equation \(0=\mathrm{d}\psi+\tfrac{1}{2}[\psi,\psi]\), then \(\operatorname{CS}(\psi)=\operatorname{CS}(\psi^{\top})\), so that the Chern–Simons \(3\)-form only detects the \(\mathfrak{g}\)-valued part of \(\psi\).

The obstruction of Chern–Simons is obtained from our main result by observing that the Gauss map of an isometric immersion \((M,g) \to \mathbb{E}^4\) is covered by a map \(F : \mathcal{SO}M \to \mathrm{SO}(4)\) which is a flat extension of the Levi-Civita connection \(\theta\) of type \((\mathrm{SO}(4),\mathrm{SO}(3))\). Since \((\mathfrak{so}(4),\mathfrak{so}(3))\) is a symmetric pair, it remains to argue that for a suitable choice of \(\langle\,\cdot\,,\cdot\,\rangle\) on \(\mathfrak{so}(4)\), the \(3\)-form \(\operatorname{CS}(\mu_{\mathrm{SO}(4)})\) represents an element of \(H^3(\mathrm{SO}(4),\mathbb{Z})\). This requires a good understanding of the third integral homology group of \(\mathrm{SO}(4)\). The relevant calculations are carried out in Section 7.

As a further application – see Proposition 6.3 – we discuss the case of space- and time-oriented Lorentzian \(3\)-manifolds that admit a global orthonormal frame. Here the Chern–Simons invariant is \(\mathbb{R}\)-valued and for appropriate choices of \(\widetilde{\mathfrak g}\) flat extensions are obtained from an isometric immersion into the Lorentzian vector space \(\mathbb R^{3,1}\) and the split-signature vector space \(\mathbb R^{2,2}\), respectively. Existence of such an immersion then leads to integrality, respectively, vanishing of the Chern–Simons invariant.

The group \(\mathrm{SO}(3)\subset \mathrm{SL}(3,\mathbb{R})\) is a strong deformation retract by Iwasawa decomposition. As a consequence of this, one also obtains an \(\mathbb{R}/\mathbb{Z}\)-valued Chern–Simons invariant for an oriented \(3\)-manifold \(M\) equipped with a torsion-free connection \(\nabla\) on its tangent bundle that preserves some volume form \(\nu\). In Theorem 6.5 we show that the vanishing of this invariant obstructs the existence of an equiaffine immersion of \((M,\nabla,\nu)\) into \(\mathbb{R}^4\), where \(\mathbb{R}^4\) is equipped with its standard flat connection and volume form.

In Section 8 we discuss two examples. In particular, applying Theorem 6.5, we show that for \(\mathbb{RP}^3\) equipped with the Levi-Civita connection \(\nabla\) and volume form \(\nu\) arising from its standard metric, there exists no global equiaffine immersion into \(\mathbb{R}^4\).

Acknowledgments

T.M. is grateful to Lukas Lewark for helpful communications.

2 Basics on the Chern–Simons \(3\)-form

We start by collecting some basic facts about Chern–Simons \(3\)-forms. We refer to [6] for additional context. We consider a Lie algebra \(\mathfrak{g}\) and study \(\mathfrak{g}\)-valued differential forms. In addition, we assume that \(\mathfrak{g}\) is endowed with a non-degenerate \(\mathfrak{g}\)-invariant, symmetric bilinear form, which we denote by \(\langle\,\cdot\,,\cdot\,\rangle\). Recall that for a simple Lie algebra \(\mathfrak{g}\) any such form has to be a multiple of the Killing form, so there is a unique such form up to scale in this case. The normalization of the form will be important in what follows, however. For a manifold \(N\), we will denote by \(\Omega^k(N,\mathfrak{g})\) the space of \(\mathfrak g\)-valued \(k\)-forms on \(N\). The main case we are interested in is that \(N\) is the total space of a principal fibre bundle and we aim at invariants defined on the base \(M\) of that bundle. For any \(N\), we have two basic operations on \(\mathfrak{g}\)-valued forms, as \[\begin{gathered} \langle\,\cdot\,,\cdot\,\rangle:\Omega^p(N,\mathfrak{g})\times\Omega^q(N,\mathfrak{g})\to \Omega^{p+q}(N) \\ [\,\,,\,]:\Omega^p(N,\mathfrak{g})\times\Omega^q(N,\mathfrak{g})\to \Omega^{p+q}(N,\mathfrak{g}). \end{gathered}\] These are characterized by the fact that for \(\alpha\in\Omega^p(N)\), \(\beta\in\Omega^q(N)\) and \(X,Y\in\mathfrak{g}\), we have \[ \langle\alpha\otimes X,\beta\otimes Y\rangle:=\langle X,Y\rangle \alpha\wedge\beta, \qquad [\alpha\otimes X,\beta\otimes Y]:=\alpha\wedge\beta\otimes[X,Y].\](2.1) The definitions immediately imply that for \(\omega\in\Omega^p(N,\mathfrak{g})\) and \(\theta\in\Omega^q(N,\mathfrak{g})\) one obtains \[\langle\omega,\theta\rangle=(-1)^{pq}\langle\theta,\omega\rangle, \qquad [\omega,\theta]=(-1)^{pq+1}[\theta,\omega],\] as well as \[\begin{gathered} \mathrm{d}\langle\omega,\theta\rangle=\langle\mathrm{d}\omega,\theta\rangle+(-1)^p\langle\omega,\mathrm{d}\theta\rangle, \\ \mathrm{d}[\omega,\theta]=[\mathrm{d}\omega,\theta]+(-1)^p[\omega,\mathrm{d}\theta]. \end{gathered}\](2.2)–(2.3) The fact that \(\langle\,\cdot\,,\cdot\,\rangle\) is \(\mathfrak{g}\)-invariant respectively the Jacobi identity for \([\ ,\ ]\) implies that for an additional form \(\tau\in\Omega^r(N,\mathfrak{g})\), we obtain \[\begin{gathered} \langle\omega,[\tau,\theta]\rangle=\langle[\omega,\tau],\theta\rangle,\\ [\omega,[\tau,\theta]]=[[\omega,\tau],\theta]+(-1)^{pr}[\tau,[\omega,\theta]]. \end{gathered}\](2.4)–(2.5)

Notice that when we write \(\Theta=\mathrm{d}\theta+\frac{1}{2}[\theta,\theta]\) we have \[\operatorname{CS}(\theta)=\langle\theta,\Theta\rangle-\tfrac{1}{6}\langle\theta,[\theta,\theta]\rangle.\] The following well-known property is one of the main motivations for the definition of the Chern–Simons \(3\)-form: \[\mathrm{d}\operatorname{CS}(\theta)=\langle\Theta,\Theta\rangle.\](2.6) The construction is natural in the sense that if \(M\) is a smooth manifold and \(\sigma :M \to N\) a smooth map, then \[ \sigma^*\operatorname{CS}(\theta)=\operatorname{CS}(\sigma^*\theta).\](2.7) Note that (2.6) in particular implies that if \(\theta\) satisfies the Maurer–Cartan equation \(\mathrm{d}\theta+\frac12[\theta,\theta]=0\), then \(\mathrm{d}\operatorname{CS}(\theta)=0\) and hence \(\operatorname{CS}(\theta)\) determines a well-defined cohomology class in \(H^3(N,\mathbb{R})\).

3 Chern–Simons invariants

The standard setting for Chern–Simons invariants associated to \((\mathfrak g,\langle\,,\,\rangle)\) uses principal connection forms on principal fibre bundles with structure group a Lie group \(G\) with Lie algebra \(\mathfrak{g}\). In some situations, it is advantageous to use a more general setting, namely to start from principal bundles whose structure group \(H\) is a Lie subgroup of a Lie group \(G\) with Lie algebra \(\mathfrak g\). Then the Lie algebra \(\mathfrak{h}\) of \(H\) naturally is a subalgebra of \(\mathfrak{g}\) and we can then restrict the adjoint representation of \(G\) to \(H\). This provides an extension of the adjoint representation of \(H\) to a representation on \(\mathfrak{g}\) that we also denote by \(\operatorname{Ad}\) if there is no risk of confusion.

In this situation, there is a natural notion of \(\mathfrak{g}\)-connections on principal fibre bundles with structure group \(H\). To formulate this, assume that \(\pi: P\to M\) is a principal \(H\)-bundle. Let \(R :P \times H \to P\) denote the principal right action and define for all \(h \in H\) the map \(R_h=R(\cdot,h) :P \to P\) and for all \(u \in P\) the map \(\iota_u=R(u,\cdot) :H \to P\).

Fixing \((\mathfrak{g},\langle\,\cdot\,,\cdot\,\rangle)\) and \(H\subset G\), we can consider the Maurer–Cartan form \(\mu_H\in\Omega^1(H,\mathfrak{h})\) and we can of course also view \(\mu_H\) as an element of \(\Omega^1(H,\mathfrak{g})\). Hence we can form the associated Chern–Simons form \(\operatorname{CS}(\mu_H)\in\Omega^3(H,\mathbb{R})\), which we will also denote by \(\operatorname{CS}^{\mathfrak{g}}(\mu_H)\) to emphasize the role of \(\mathfrak{g}\). Since \(\mu_H\) satisfies the Maurer–Cartan equation, this form is closed by (2.6) and hence determines a cohomology class \([\operatorname{CS}^{\mathfrak{g}}(\mu_H)]\in H^3(H,\mathbb{R})\).

Explicitly, \(\operatorname{CS}(\mu_H)\) is the left-invariant \(3\)-form on \(H\), which is induced by the trilinear map \[\mathfrak{h}^3\to\mathbb{R}, \qquad (X,Y,Z)\mapsto -\frac16\langle X,[Y,Z]\rangle\] which is complete alternating by invariance of \(\langle\,\cdot\,,\cdot\,\rangle\). If \(\mathfrak{h}\) is simple, then the restriction of \(\langle\,\cdot\,,\cdot\,\rangle\) to \(\mathfrak{h}\) has to be a multiple of the Killing form, and hence \(\operatorname{CS}^{\mathfrak{g}}(\mu_H)\) is a multiple of the so-called Cartan \(3\)-form on \(H\). It is well-known that for a compact simple Lie group \(H\), the cohomology class of the Cartan \(3\)-forms spans \(H^3(H,\mathbb{R})\cong\mathbb{R}\). We now have:

The proof of Proposition 3.1 relies on the identity (3.2) below which is standard in the case of principal connections (see for instance [6]). The case of \(\mathfrak{g}\)-valued connection forms is quite similar. For the convenience of the reader we include a proof in our more general setting:

4 Examples of Chern–Simons invariants

Recall that any orientable \(3\)-manifold is parallelizable and hence admits a global smooth frame for the tangent bundle.

(1) Taking \(H=G=\mathrm{SO}(3)\), this recovers the original definition of Chern–Simons by using the Levi-Civita connection on oriented Riemannian \(3\)-manifolds. Starting from a global frame of the tangent bundle \(TM\), we can apply Gram-Schmidt to obtain a global orthonormal frame, which shows that the orthonormal frame bundle admits global smooth sections. Since \(H^3(G,\mathbb{R})\cong\mathbb{R}\) we immediately conclude from Proposition 3.1 that we obtain an invariant with values in \(\mathbb{R}/\mathbb{Z}\) provided that we normalize \(\langle\,\cdot\,,\cdot\,\rangle\) in such a way that \(\int_{\mathrm{SO}(3)}\mu_{\mathrm{SO}(3)}\in\mathbb{Z}\) and the best choice is to ensure that \(\int_{\mathrm{SO}(3)}\mu_{\mathrm{SO}(3)}=\pm 1\). Such a normalization is computed explicitly in Section 7.

(2) Take \(H=G=\mathrm{SO}_0(2,1)\) and the Levi-Civita connection on a space- and time-oriented Lorentzian \(3\)-manifold. In this case, we have to assume in addition that there is a global orthonormal frame for the given Lorentzian metric, i.e. that the orthonormal frame bundle admits a global smooth section. Since the maximal compact subgroup of \(G\) is contained in \(\mathrm{S}(\mathrm{O}(2)\times\mathrm{O}(1))\), we get \(H^3(G,\mathbb{R})=\{0\}\) in this case. Hence regardless of the normalization of \(\langle\,\cdot\,,\cdot\,\rangle\), we get an \(\mathbb{R}\)-valued invariant (assuming existence of a global orthonormal frame). As we shall see below, the normalization of \(\langle\,\cdot\,,\cdot\,\rangle\) still can be relevant here, since there are integrality results for the invariant on manifolds that admit certain isometric immersions, see Proposition 6.3 below.

(3) Take \(H=G=\mathrm{SL}(3,\mathbb{R})\) and consider oriented \(3\)-manifolds \(M\) endowed with a fixed volume form \(\nu\) and a linear connection \(\nabla\) on \(TM\) preserving \(\nu\). Then we can apply our construction to the principal connection induced by \(\nabla\) on the \(\mathrm{SL}(3,\mathbb{R})\)-frame bundle of \(M\) defined by \(\nu\). The inclusion \(\mathrm{SO}(3)\to \mathrm{SL}(3,\mathbb{R})\) induces an isomorphism in cohomology, so \(H^3(H,\mathbb{R})\cong\mathbb{Z}\). Choosing \(\langle\,\cdot\,,\cdot\,\rangle\) appropriately, we obtain an \(\mathbb{R}/\mathbb{Z}\)-valued invariant for such connections.

(4) An example with \(H\neq G\) is provided by the Burns-Epstein invariant introduced in [1]. Here the setting is that \(M\) is a compact oriented \(3\)-manifold endowed with a CR-structure, i.e. a contact distribution \(C\subset TM\) which is endowed with an almost complex structure. It is a classical result due to E. Cartan, see [3], that \(M\) admits a canonical \(\mathfrak{g}\)-connection with \(\mathfrak{g}=\mathfrak{su}(2,1)\) on a principal fibre bundle constructed from the CR structure. Indeed, this is a Cartan connection in modern terminology.

The structure group \(H\) of the canonical principal bundle is a (parabolic) subgroup of \(G=\mathrm{PSU}(2,1)\). The details on \(H\) are not very important here, it comes from the stabilizer in \(\mathrm{SU}(2,1)\) of a null line in \(\mathbb C^3\). It turns out that \(H\) is isomorphic to a semi-direct product of \(\mathrm{U}(1)\) and the complex Heisenberg group of real-dimension \(3\), which in particular implies that \(H^3(H,\mathbb{R})=\{0\}\).

To obtain an invariant, one has to assume that the \(H\)-principal bundle associated to the CR structure admits a global section, which by orientability of \(M\) turns out to be equivalent to triviality of the CR subbundle \(C\subset TM\). Equivalently, this can be expressed as existence of a global CR vector field on \(M\). Since \(\mathfrak{g}\) is simple, the form \(\langle\,\cdot\,,\cdot\,\rangle\) has to be a non-zero multiple of the Killing form, and for each choice of such a form, our construction leads to a real-valued CR invariant. Notice that in this case the restriction of \(\langle\,\cdot\,,\cdot\,\rangle\) to \(\mathfrak{h}\) is degenerate with null-space the nilradical of \(\mathfrak{h}\).

(5) The construction of canonical principal bundles and (Cartan) connections from (4) is a special case of the general constructions for parabolic geometries, see [2]. In particular, there are two more cases of structures on \(3\)-manifolds that have an underlying contact structure. One of those are Legendrian contact structures for which the additional ingredient to a contact distribution \(C\subset TM\) is a decomposition \(C=E\oplus F\) as the direct sum of two line subbundles (which are automatically Legendrian). Here \(\mathfrak{g}=\mathfrak{sl}(3,\mathbb{R})\) and \(H\) comes from the subgroup of upper triangular matrices in \(\mathrm{SL}(3,\mathbb{R})\), so \(H^3(H,\mathbb{R})=\{0\}\). To obtain an invariant, one again has to assume that \(M\) is compact and oriented and that the Legendrian subbundles \(E\) and \(F\) are trivial, and then one obtains an \(\mathbb{R}\)-valued invariant.

The other example are so-called contact projective structures, see also [5]. Here the additional structure is given by a family of curves tangent to the contact distribution \(C\subset TM\) with one curve through each point in each direction (in \(C\)), which can be realized as geodesics of a linear connection. For this example, \(\mathfrak{g}=\mathfrak{sp}(4,\mathbb{R})\) and \(H\) comes from a subgroup of \(\mathrm{Sp}(4,\mathbb{R})\) which is a semi-direct product of \(\mathrm{Sp}(2,\mathbb{R})\cong \mathrm{SL}(2,\mathbb{R})\) with a \(3\)-dimensional real Heisenberg group. Hence \(H^3(H,\mathbb{R})=\{0\}\) and assuming triviality of the canonical principal bundle over a compact oriented \(3\)-manifold, one thus obtains a real-valued invariant for any choice of \(\langle\,\cdot\,,\cdot\,\rangle\).

5 Partial blindness for flat Lie algebra valued forms

It is natural to try to find conditions on the pair \((\pi :P \to M,\theta)\) which imply vanishing of the associated Chern–Simons invariant. We will achieve this in Section 6 below. Our result crucially relies on a certain algebraic feature of the Chern–Simons \(3\)-form that we will refer to as partial blindness for flat Lie algebra valued forms.

Here our basic setup is that we study Chern–Simons invariants associated to \((\mathfrak{g},\langle\,\cdot\,,\cdot\,\rangle)\) via a realization of \(\mathfrak{g}\) as a Lie subalgebra of a bigger Lie algebra \(\tilde{\mathfrak{g}}\) in such a way that \(\langle\,\cdot\,,\cdot\,\rangle\) is the restriction of an invariant form on \(\tilde{\mathfrak{g}}\) which we denote by the same symbol. In particular we do assume that the restriction to \(\mathfrak{g}\) is non-degenerate here. (Note that this is automatically satisfied if \(\mathfrak{g}\) is simple and the restriction is non-zero.) Then we can form the orthogonal space \(\mathfrak{g}^\perp\subset\mathfrak{\tilde{g}}\) which by our assumption is complementary to \(\mathfrak{g}\), so \(\mathfrak{\tilde{g}}=\mathfrak{g}\oplus\mathfrak{g}^\perp\) as a vector space. But since \(\langle\,\cdot\,,\cdot\,\rangle\) is clearly \(\mathfrak{g}\)-invariant, also \(\mathfrak g^\perp\subset\mathfrak{\tilde{g}}\) is a \(\mathfrak{g}\)-invariant subspace, so \(\mathfrak{\tilde{g}}=\mathfrak{g}\oplus\mathfrak{g}^\perp\) as a representation of \(\mathfrak{g}\). In particular, this implies that \([\mathfrak{g},\mathfrak{g}^\perp]\subset\mathfrak{g}^\perp\). An important special case is that the complement \(\mathfrak{g}^\perp\) makes \((\mathfrak{\tilde{g}},\mathfrak{g})\) into a symmetric pair, i.e., that in addition we have \([\mathfrak{g}^\perp,\mathfrak{g}^\perp]\subset\mathfrak{g}\).

Now any \(\mathfrak{\tilde{g}}\)-valued form \(\omega\in\Omega^k(N,\mathfrak{\tilde{g}})\) on a manifold \(N\) decomposes accordingly as \[\omega=\omega^\top+\omega^\perp\] with \(\omega^\top\in\Omega^k(N,\mathfrak{g})\) and \(\omega^{\perp}\in\Omega^k(N,\mathfrak{g}^\perp)\).

We can now state a crucial technical lemma:

As a first application of Lemma 5.1 we assume that we have an inclusion \(i:G\to\tilde G\) of groups corresponding to \(\mathfrak{g}\subset\tilde{\mathfrak{g}}\) and get a result of integrality of the Chern–Simons forms associated to the Maurer–Cartan forms.

There is a conceptual way to obtain compatible invariant bilinear forms as we need them here. Assume that \(\tilde{\mathfrak{g}}\) is realized as a subalgebra of \(\mathfrak{gl}(n,\mathbb{R})\) for some \(n\). Then it is well-known that the trace form \(\langle X,Y\rangle:=\operatorname{tr}(XY)\) defines a \(\mathrm{GL}(n,\mathbb{R})\)-invariant bilinear form on \(\mathfrak{gl}(n,\mathbb{R})\). Of course, the restriction of \(\langle\,\cdot\,,\cdot\,\rangle\) to \(\tilde{\mathfrak{g}}\) is the \(\tilde G\)-invariant for the subgroup \(\tilde G\subset \mathrm{GL}(n,\mathbb{R})\) corresponding to the Lie subalgebra \(\tilde{\mathfrak{g}}\). If \(\tilde{\mathfrak g}\) is simple and the restriction is non-zero, then it has to be a multiple of the Killing form and hence is non-degenerate. Of course, we can restrict further to \(\mathfrak{g}\subset\tilde{\mathfrak{g}}\) and apply the same argument if \(\mathfrak{g}\) is also simple.

Even better, for \(m>n\), we can decompose \(\mathbb{R}^m=\mathbb{R}^{m-n}\oplus\mathbb{R}^n\) and then include \(\mathrm{GL}(n,\mathbb{R})\subset \mathrm{GL}(m,\mathbb{R})\) as the maps that send \(\mathbb{R}^n\) to itself and are the identity on the complementary subspace \(\mathbb{R}^{m-n}\). For the infinitesimal inclusion \(\mathfrak{gl}(n,\mathbb{R})\to\mathfrak{gl}(m,\mathbb{R})\) the trace form on \(\mathfrak{gl}(m,\mathbb{R})\) evidently restricts to the trace form on \(\mathfrak{gl}(n,\mathbb R)\). This allows us to also compare algebras of matrices of different sizes.

It is easy to write an explicit formula for Chern–Simons forms on subalgebras of \(\mathfrak{gl}(n,\mathbb{R})\) computed with respect to the trace form. One realizes a \(1\)-form \(\theta\) on \(N\) with values in such a subalgebra as a matrix \((\theta^i_j)_{i,j=1}^n\) of real-valued \(1\)-forms \(\theta^i_j\in\Omega^1(N)\). Then of course, we get \(\mathrm{d}\theta=(\mathrm{d}\theta^i_j)_{i,j=1}^n\), and employing the summation convention, we can write \([\theta,\theta]=(2\theta^i_k\wedge\theta^k_j)_{i,j=1}^n\). By definition of the Chern–Simons form, this implies that we get (still using the summation convention) \[\operatorname{CS}(\theta)=\theta^i_j\wedge\mathrm{d}\theta^j_i+\tfrac{2}{3}\theta^i_j\wedge\theta^j_k\wedge\theta^k_i.\] It is usual in Chern–Simons theory to write this as \(\operatorname{tr}(\theta\wedge \mathrm{d}\theta+\tfrac{2}{3}\theta\wedge\theta\wedge\theta)\). Using this, we can prove a result on the Lie algebras we will need in the further developments:

The explicit normalizations we will use are computed in Section 7 below.

6 Flat extensions of principal connections

We continue to work in the setting of an inclusion \(i:G\to\tilde G\) of a Lie subgroup and an invariant bilinear form \(\langle\,\cdot\,,\cdot\,\rangle\) on \(\tilde{\mathfrak{g}}\) whose restriction to \(\mathfrak{g}\) is non-degenerate. All Chern–Simons forms will be formed with respect to this bilinear form from now on. In this setting recall from 1 our key concept of a flat extension:

Thus we see that in the cases where we obtain \(\mathbb{R}/\mathbb{Z}\)-valued invariants, existence of a flat extension implies vanishing of the invariant. In the case of \(\mathbb{R}\)-valued invariants, the consequence of flat extensions depends on the topology of \(\widetilde{G}\). In case that \(H^3(\widetilde{G},\mathbb{R})=\{0\}\) (or more generally \([\mu_{\widetilde{G}}]=0\)), we again get vanishing of the invariant. However, if \(0\neq [\mu_{\widetilde{G}}]\in H^3(\widetilde{G},\mathbb{Z})\), then we can only conclude that the invariant is an integer.

6.1 Geometric construction of Riemannian flat extensions

The basic example of a flat extension comes from the original article of Chern–Simons. Suppose that \((M,g)\) is an oriented Riemannian \(3\)-manifold, so \(G=H=\mathrm{SO}(3)\). Then we choose \(\tilde G=\mathrm{SO}(4)\), which contains \(G\) as a subgroup in the obvious way, i.e. as matrices of the form \[\begin{pmatrix} 1 & 0 \\ 0 & A \end{pmatrix}.\] One immediately verifies that for the Lie algebras this implies that \((\widetilde{\mathfrak g},\mathfrak{g})\) is a symmetric pair.

Now \(\tilde G\) acts transitively on the space of oriented \(3\)-planes in the inner product space \(\mathbb{R}^4\) and for this action, \(G\) is the stabilizer of the oriented subspace \(\mathbb{R}^3\subset\mathbb{R}^4\). Hence we can identify the symmetric space \(\mathrm{SO}(4)/\mathrm{SO}(3)\) as the space of all oriented \(3\)-planes in \(\mathbb{R}^4\). An oriented \(3\)-plane in \(\mathbb{R}^4\) is of course uniquely determined by its positive unit normal, which leads to the more common identification \(\mathrm{SO}(4)/\mathrm{SO}(3)\cong S^3\).

As the \(\mathrm{SO}(4)\)-invariant inner product on \(\widetilde{\mathfrak g}\), we use \(\tfrac{1}{16\pi^2}\) times the trace form. Proposition 7.1 in Section 7 below shows that this is a minimal choice for which we get \([\operatorname{CS}(\mu_{\mathrm{SO}(4)})]\in H^3(\mathrm{SO}(4),\mathbb{Z})\). By Lemma 5.2, the restriction of this form to \(\mathfrak{so}(3)\) has the property that \([\operatorname{CS}(\mu_{\mathrm{SO}(3)})]\in H^3(\mathrm{SO}(3),\mathbb{Z})\). More precisely, Proposition 7.1 actually shows that \(\int_{\mathrm{SO}(3)}\operatorname{CS}(\mu_{\mathrm{SO}(3)})=1\), so this is an optimal choice of normalization for obtaining an invariant with values in \(\mathbb{R}/\mathbb{Z}\).

6.2 Lorentzian flat extensions

The line of argument of Section 6.1 carries over directly to the Lorentzian case, but things become more interesting here. On the one hand, so far no normalization of the invariant bilinear form was needed in this case, but this becomes important now. On the other hand, we have now two possibilities for isometric immersions, namely either into \(\mathbb R^{3,1}\) (with positive normal) or into \(\mathbb R^{2,2}\) (with negative normal). As we shall see, these have different consequences for the invariant. The corresponding choices for \(\tilde G\) are \(\mathrm{SO}_0(3,1)\) and \(\mathrm{SO}_0(2,2)\) respectively, with the obvious inclusion of \(G=\mathrm{SO}_0(2,1)\) in both cases. In both cases, one obtains a symmetric pair \((\widetilde{\mathfrak g},\mathfrak{g})\). The homogeneous space \(\mathrm{SO}_0(3,1)/\mathrm{SO}_0(2,1)\) can be interpreted as the space of all space- and time-oriented Lorentzian linear subspaces \(V\subset\mathbb{R}^{3,1}\). For any such subspace there is a unique oriented (positive) unit normal. Similarly, \(\mathrm{SO}_0(2,2)/\mathrm{SO}_0(2,1)\) is the space of all space- and time-oriented Lorentzian linear subspaces \(V\subset\mathbb{R}^{2,2}\) and such a space determines a unique (negative) oriented unit normal.

The difference between the two cases comes from the structure of \(H^3(\tilde G,\mathbb R)\). For \(\mathrm{SO}_0(3,1)\), the maximal compact subgroup is isomorphic to \(\mathrm{SO}(3)\), so the third cohomology is non-trivial here. For \(\mathrm{SO}_0(2,2)\), the maximal compact subgroup is two dimensional and hence the third cohomology vanishes identically. Thus the appropriate normalization for our purposes is to start with an invariant non-degenerate symmetric bilinear form on \(\mathfrak{so}(3,1)\) normalized in such a way that \([\operatorname{CS}(\mu_{\mathrm{SO}_0(3,1)})]\in H^3(\mathrm{SO}_0(3,1),\mathbb{Z})\) for the resulting normalization. As we have seen in Section 6.1 above, \(\tfrac{1}{16\pi^2}\) times the trace form leads to an optimal normalization for \(\mathrm{SO}(3)\). By the Iwasawa decomposition, \(\mathrm{SO}(3)\) is a deformation retract of \(\mathrm{SO}_0(3,1)\), so we can use the same normalization here. Now the proof of Proposition 6.2 generalizes in a straightforward way to show

The interpretation of \(F^*(\mu_{\tilde G}^\perp)\) for a flat extension \(F\) is completely parallel to the Riemannian case discussed in 17.

6.3 Equiaffine immersions and flat extensions

We next discuss the case of volume preserving affine connections on \(3\)-manifolds, so \(G=H=\mathrm{SL}(3,\mathbb{R})\). Similarly to the pseudo-Riemannian case, we consider \(\widetilde{G}:=\mathrm{SL}(4,\mathbb{R})\) and the obvious inclusion of \(\mathrm{SL}(3,\mathbb{R})\) here and the flat extension will be obtained from a Gauss map. The details will be quite a bit more involved, though, in particular \(\tilde G/G\) has dimension \(7\) and is not a symmetric space. Indeed, on the level of Lie algebras, the inclusion \(\mathfrak g:=\mathfrak{sl}(3,\mathbb{R})\hookrightarrow \mathfrak{sl}(4,\mathbb{R})=:\widetilde{\mathfrak g}\) corresponds to a block decomposition of elements of \(\widetilde{\mathfrak g}\) of the form \[\begin{pmatrix} a & Z \\ X & A-\tfrac{a}{3}\mathbb I\end{pmatrix}\] with \(A\in\mathfrak{g}\), \(a\in\mathbb{R}\), \(X\in\mathbb{R}^3\) and \(Z\in\mathbb{R}^{3*}\). Hence we conclude that \(\mathfrak{g}^\perp=\mathbb R\oplus\mathbb{R}^3\oplus\mathbb{R}^{3*}\) as a representation of \(\mathfrak{g}\), and the component \(\mathfrak{g}^\perp\times\mathfrak{g}^\perp\to\mathfrak{g}^\perp\) of the Lie bracket of \(\widetilde{\mathfrak g}\) is (in obvious notation) given by \[\begin{gathered} [(a_1,X_1,Z_1),(a_2,X_2,Z_2)]\\ =(Z_1X_2-Z_2X_1,\frac43(-a_1X_2+a_2X_1),\frac43(a_1Z_2-a_2Z_1)). \end{gathered}\](6.2)

Let us first give an appropriate interpretation of the homogeneous space \(\tilde G/G\). Consider the set of all pairs \((V,\ell)\), where \(V\subset\mathbb{R}^4\) is an oriented linear subspace of dimension \(3\) endowed with a volume element and \(\ell\subset\mathbb{R}^4\) is a \(1\)-dimensional linear subspace complementary to \(V\). Then \(\mathrm{SL}(4,\mathbb{R})\) acts on this space by \(A\cdot (V,\ell):=(A(V),A(\ell))\). Starting from the natural base point given by \(V=\mathbb{R}^3\) with the orientation and volume element induced by the standard basis \(\{e_1,e_2,e_3\}\) and \(\ell=\mathbb{R}\cdot e_4\), linear algebra shows that the action of \(\mathrm{SL}(4,\mathbb{R})\) is transitive and that the stabilizer of the base point is \(\mathrm{SL}(3,\mathbb{R})\). Hence we may identify \(\tilde G/G\) with the space of all such pairs.

Now let us assume that we have given an oriented \(3\)-manifold \(M\). Suppose that \(i:M\to\mathbb{R}^4\) is an immersion and that in addition we choose a transversal, i.e. for each \(x\in M\), we choose a line \(\ell(x)\subset\mathbb R^4\), which is complementary to the \(3\)-dimensional subspace \(T_xi(T_xM)\subset\mathbb R^4=T_{i(x)}\mathbb{R}^4\). We assume this choice to be smooth in the obvious sense, i.e. locally around each point the lines can be spanned by a smooth vector field along \(i\). This shows that \(i\) and \(\ell\) determine a smooth Gauss map \(f: M\to\tilde G/G\), which sends each \(x\in M\) to the pair \((T_xi(T_xM),\ell(x))\).

Via \(i\) and \(\ell\), any tangent vector at a point \(i(x)\) decomposes uniquely into a component in \(\ell(x)\) and a component in \(T_xi(T_xM)\). Via this decomposition, the flat connection on \(\mathbb{R}^4\) induces a linear connection \(\nabla^\top\) on \(TM\) as well as a connection \(\nabla^\perp\) on the trivial line bundle \(M\times\mathbb{R}\). It is easy to understand when \(\nabla^\top\) preserves a volume form.

Now we introduce the classical concept of an equiaffine immersion, as in [10].

As the invariant bilinear form on \(\widetilde{\mathfrak g}=\mathfrak{sl}(4,\mathbb{R})\), we again use \(\tfrac{1}{16\pi^2}\) times the trace form. In view of Proposition 5.3 the considerations in Section 6.1 show that this leads to \([\operatorname{CS}(\mu_{\mathrm{SL}(4,\mathbb{R})})]\in H^3(\mathrm{SL}(4,\mathbb{R}),\mathbb{Z})\) as well as \(\int_{\mathrm{SO}(3)}\operatorname{CS}(\mu_{\mathrm{SL}(3,\mathbb{R})})=1\).

7 Normalization

We can apply the construction of the Riemannian Gauss map from Section 6.1 (in general dimensions) to the inclusion \(i:S^n\to\mathbb R^{n+1}\) of the standard sphere. Lifting this to the oriented orthonormal frame bundle \(\mathcal{SO}S^n\) one obtains an isomorphism \(F:\mathcal{SO}S^n\to \mathrm{SO}(n+1)\) covering the diffeomorphism \(S^n\to\mathrm{SO}(n+1)/\mathrm{SO}(n)\) discussed in Section 6.1. This is most easily interpreted as taking the map \(p:\mathrm{SO}(n+1)\to S^n\) defined by mapping \(A\in\mathrm{SO}(n+1)\) to \(A(e_1)\), i.e. the first column vector of \(A\). This is the projection of the oriented orthonormal frame bundle, since the remaining columns of \(A\) are a positively oriented orthonormal basis for \(A(e_1)^\perp=T_{A(e_1)}S^n\). The considerations in Section 6.1 also show that for the Maurer-Cartan form \(\mu_{n+1}\) of \(\mathrm{SO}(n+1)\), \(\mu_{n+1}^\top\) is the connection form of the Levi-Civita connection of the round metric on \(S^n\), while \(\mu_{n+1}^\perp\) represents the second fundamental form.

8 Examples

8.1 Lorentzian metrics

We compute the Chern–Simons invariant for a \(1\)-parameter family of left-invariant Lorentzian metrics on \(\mathrm{SU}(2)\). Under the identification \(\mathrm{SU}(2)\simeq S^3\subset \mathbb{R}^4\) given by \[\begin{pmatrix} x_1+\mathrm{i}x_2 & -x_3+\mathrm{i}x_4 \\ x_3+\mathrm{i}x_4 & x_1-\mathrm{i}x_2 \end{pmatrix} \mapsto \begin{pmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{pmatrix}=x\] for all \(x \in \mathbb{R}^4\) with \(|x|=1\), the Maurer–Cartan form of \(\mathrm{SU}(2)\) can be written as \[\mu:=\mu_{\mathrm{SU}(2)}=\begin{pmatrix} -\mathrm{i}\kappa & -\xi+\mathrm{i}\rho \\ \xi+\mathrm{i}\rho & \mathrm{i}\kappa \end{pmatrix},\] where \(\xi,\rho,\kappa\) are given by (7.3). The Maurer–Cartan equation \(\mathrm{d}\mu+\tfrac{1}{2}[\mu,\mu]=0\) is equivalent to the structure equations \[\mathrm{d}\xi=-2\rho\wedge\kappa, \qquad \mathrm{d}\rho=-2\kappa\wedge\xi, \qquad \mathrm{d}\kappa=-2\xi\wedge\rho.\] For \(\lambda \in \mathbb{R}\setminus\{0\}\) consider the Lorentzian metric \(g_{\lambda}=\xi^2+\rho^2-\lambda^2\kappa^2\). This can be viewed as a natural Lorentzian analog of a Berger sphere or of a pseudo-Hermitian structure on the CR-manifold \(\mathrm{SU}(2)\) induced by a left-invariant contact form. The Levi-Civita connection form \(\theta_{\lambda}\) of \(g_{\lambda}\) – when pulled back with respect to the section \(\sigma_{\lambda} : \mathrm{SU}(2) \to \mathcal{SO}_0\mathrm{SU}(2)\) corresponding to \((\xi,\rho,\lambda\kappa)\) – becomes \[\sigma^*_{\lambda}\theta_{\lambda}= \begin{pmatrix} 0 & -(\lambda^2+2)\kappa & -\lambda\rho \\ (\lambda^2+2)\kappa & 0 & \lambda\xi \\ -\lambda\rho & \lambda\xi & 0 \end{pmatrix}.\] Using the normalisation of Section 7, we compute for the Chern–Simons \(3\)-form \[\sigma^*_{\lambda}\operatorname{CS}(\theta_{\lambda})=\left(\frac{8}{16\pi^2}\right)(\lambda^4+2\lambda^2+2)\,\xi\wedge\rho\wedge\kappa.\] Consequently, (7.5) gives \[\int_{\mathrm{SU}(2)}\sigma^*_{\lambda}\operatorname{CS}(\theta_{\lambda})=\lambda^4+2\lambda^2+2>0,\] so that Proposition 6.3 implies that there is no isometric immersion of \(g_{\lambda}\) into \(\mathbb{R}^{2,2}\) for all \(\lambda \in \mathbb{R}\setminus\{0\}\) and that there is no isometric immersion of \(g_{\lambda}\) into \(\mathbb{R}^{3,1}\) for all \(\lambda \in \mathbb{R}\setminus\{0\}\) such that \(\lambda^4+2\lambda^2+2 \notin \mathbb{Z}\).

8.2 Equiaffine immersion

Real projective \(3\)-space \(\mathbb{RP}^3\) equipped with its standard metric \(g_{\mathrm{St}}\) is isometric to \(\mathrm{SO}(3)\) equipped with the metric \(g=\tfrac{1}{4}\left((\omega_1)^2+(\omega_2)^2+\psi^2\right)\). The Levi-Civita connection form \(\theta\) of \(g\) – when pulled back with respect to the section \(\sigma : \mathrm{SO}(3) \to \mathcal{SO}\mathrm{SO}(3)\) corresponding to \(\tfrac{1}{2}(\omega_1,\omega_2,\psi)\) – becomes \[\sigma^*\theta=\frac{1}{2}\begin{pmatrix} 0 & -\psi & \omega_2 \\ \psi & 0 & -\omega_1 \\ -\omega_2 & \omega_1 & 0 \end{pmatrix}.\] Using the normalisation of Section 7, we compute for the Chern–Simons \(3\)-form \(\sigma^*\operatorname{CS}(\theta)=\tfrac{1}{16\pi^2}\,\omega_1\wedge\omega_2\wedge\psi\) so that \[c_{\sigma}=\int_{\mathrm{SO}(3)}\sigma^*\operatorname{CS}(\theta)=\frac{1}{2} \notin \mathbb{Z}.\] Proposition 6.2 implies that there is no isometric immersion of \((\mathbb{RP}^3,g_{\mathrm{St}})\) into \(\mathbb{E}^4\), as previously observed by Chern–Simons [4]. We can however say a bit more. Let \(\nabla\) denote the Levi-Civita connection of \(g_{\mathrm{St}}\) on \(T\mathbb{RP}^3\) and \(\nu\) the \(\nabla\)-parallel volume form on \(\mathbb{RP}^3\) corresponding to \(\omega_1\wedge\omega_2\wedge\psi\). Since the normalisation of \(\langle\,\cdot\,,\cdot\,\rangle\) for \(\mathrm{SL}(3,\mathbb{R})\) is the same as for \(\mathrm{SO}(3)\), our Theorem 6.5 implies:

Funding

This research was funded in part by the Austrian Science Fund (FWF): 10.55776 / P33559. For open access purposes, the authors have applied a CC BY public copyright license to any author-accepted manuscript version arising from this submission. This article is based upon work from COST Action CaLISTA CA21109 supported by COST (European Cooperation in Science and Technology). https://www.cost.eu. The authors also would like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the programme Twistor theory, where work on this paper was undertaken. This work was supported by EPSRC grant EP/Z000580/1. T.M. was partially supported by the DFG priority programme “Geometry at infinity” SPP 2026: ME 4899/1-2.

References