Mathematics

A. L. Onishchik

Connections Without Curvature and the de Rham Theorem

(Presented by Academician P. S. Aleksandrov on 13 VI 1964)

In the paper \((^1)\) we proved Theorem 3, which relates the set of one-dimensional cohomology \(H^1(X,\mathfrak{G})\) of a manifold \(X\) with coefficients in an arbitrary Lie group \(\mathfrak{G}\) (or, equivalently, the set of locally constant fibrations with base \(X\) and group \(\mathfrak{G}\)) to differential forms on the manifold \(X\). In the present paper we refine this theorem, at the same time generalizing (in the one-dimensional case) de Rham’s theorems on the existence and cohomology of forms with prescribed periods. The role of closed forms here is played by connections with zero curvature in a fiber space; the role of periods, by the holonomy groups of these connections. In conclusion, a complex-analytic analogue of the theory is considered, and, in particular, results of Röhrl \((^2)\) concerning the Riemann–Hilbert problem are obtained.

1. Let \(X\) be a smooth manifold, \(\mathfrak{G}\) a Lie group, and \(P\) a principal fiber space with base \(X\) and group \(\mathfrak{G}\). Suppose that an infinitesimal connection with zero curvature is given in the fibration \(P\). Fix a point \(x_0 \in X\) and a point \(b_0 \in P\) lying in the fiber over \(x_0\). Then to every piecewise-smooth closed path \(l\) on \(X\), issuing from \(x_0\), there corresponds an element \(g \in \mathfrak{G}\) of the holonomy group. If \(l\) is homotopic to zero, then \(g=e\), since for a connection with zero curvature the restricted holonomy group is trivial \((^3)\). Therefore we obtain a homomorphism \(\omega:\pi_1(X)\to \mathfrak{G}\), where \(\pi_1(X)\) is the fundamental group of the manifold \(X\) referred to the point \(x_0\). We shall call the homomorphism \(\omega\) the holonomy homomorphism of our connection. Homomorphisms \(\omega_1,\omega_2:\pi_1(X)\to\mathfrak{G}\) are called conjugate if
   \[ \omega_2(h)=g_0\omega_1(h)g_0^{-1}\quad (h\in\pi_1(X)) \]
   for some \(g_0\in\mathfrak{G}\). When the point \(b_0\) in the fiber over \(x_0\) is changed, the holonomy homomorphism is replaced by a conjugate one.

We set ourselves the following problems:

1) To determine in what case the holonomy homomorphisms corresponding to two connections without curvature are conjugate to one another.

2) To determine which homomorphisms \(\pi_1(X)\to\mathfrak{G}\) can serve as holonomy homomorphisms for connections with zero curvature in \(P\).

If by \(Z(P)\) we denote the set of all connections with zero curvature in \(P\), and by \(\Omega(X,\mathfrak{G})\) the set of classes of conjugate homomorphisms \(\pi_1(X)\to\mathfrak{G}\), then there arises a mapping
\[ \mu:\quad Z(P)\to \Omega(X,\mathfrak{G}), \]
which we must study.

Consider the case in which the fibration \(P\) is trivial. Then a connection is given by a differential \(1\)-form \(\alpha\) on \(X\) with values in the Lie algebra of the group \(\mathfrak{G}\). The determination of paths horizontal in the sense of the connection \(\alpha\) is carried out by means of the differential equation
\[ f^{-1}df=\alpha, \tag{1} \]
where \(f:X\to\mathfrak{G}\) is an unknown smooth function. The condition for the triviality of the curvature is expressed by the equality
\[ d\alpha+\frac{1}{2}[\alpha,\alpha]=0. \tag{2} \]

This equality is precisely the condition for complete integrability of equation (1). The holonomy group of the connection \(\alpha\) in this case is called the monodromy group of equation (1). The corresponding homomorphism \(\omega:\pi_1(X)\to \mathfrak G\) will be called the monodromy homomorphism.

Let now \(\mathfrak G=R\), the additive group of real numbers. Then \(\alpha\) is an ordinary 1-form on \(X\). Condition (2) means that \(d\alpha=0\). It is easy to see that for \(l\in\pi_1(X)\) we have \(\omega(l)=\int_l\alpha\). According to de Rham’s theorem, two closed forms have the same periods if and only if they are cohomologous. Further, there always exists a closed form with arbitrary periods. Thus, in this case de Rham’s theorem gives a solution of problems (1) and (2). Therefore the theorem proved below, Theorem 1, may be regarded as a natural generalization of de Rham’s theorem.

1. Let \(\{U_i\}\) be a coordinate covering of the manifold \(X\) for the bundle \(P\), and let \(P\) be determined by transition functions
   \(p_{ij}:U_i\cap U_j\to \mathfrak G\). Denote by \(\operatorname{Int} P\) the bundle with base \(X\) and fiber \(\mathfrak G\) which, in the covering \(\{U_i\}\), is determined by the transition functions \(A_{p_{ij}}\), where \(A_g\) is the automorphism of the group \(\mathfrak G\) acting by the formula \(A_g(x)=gxg^{-1}\). Then \(\operatorname{Int} P\) is a group bundle. Therefore the set \(D(\operatorname{Int} P)\) of all its smooth sections is a group. This group acts on the bundle \(P\) by means of “left shifts” \(L_f\bigl(f\in D(\operatorname{Int} P)\bigr)\). It can be shown that in this way one obtains the group of all automorphisms of the principal bundle \(P\) that carry each fiber into itself. Hence there arises a group of transformations \(L_f^*\) of the set of all infinitesimal connections in the bundle \(P\), and the set \(Z(P)\) is carried into itself. Denote by \(H(P)\) the set of orbits of the group \(D(\operatorname{Int} P)\) in \(Z(P)\). It is easy to see that the holonomy homomorphisms corresponding to two connections lying in the same orbit are conjugate to one another. Therefore a mapping arises

\[ \bar\mu:\ H(P)\to \Omega(X,\mathfrak G). \]

Let us also note that every homomorphism \(\pi_1(X)\to \mathfrak G\) induces a principal bundle with base \(X\) and group \(\mathfrak G\), having locally constant transition functions (a locally constant fiber bundle). More precisely, there is a one-to-one mapping \(q\) from the set \(\Omega(X,\mathfrak G)\) onto \(H^1(X,\mathfrak G)\) \((^4)\).

Theorem 1. The mapping \(\bar\mu\) is one-to-one, and its image consists of all classes of homomorphisms \(\pi_1(X)\to \mathfrak G\) for which the corresponding fiber bundles are differentiably equivalent to the bundle \(P\).

Proof. First of all, it is not difficult to show that if the bundle \(P\) has an infinitesimal connection without curvature, then its transition functions \(p_{ij}\) can be chosen locally constant. A connection in \(P\) is given by a system of 1-forms \(\alpha_i\) on \(U_i\) with values in \(G\), satisfying on \(U_i\cap U_j\) the relation

\[ \alpha_j=\operatorname{Ad}(p_{ij}^{-1})\alpha_i+p_{ij}^{-1}dp_{ij}. \tag{3} \]

Consider the bundle \(\operatorname{Ad} P\) with base \(X\), fiber \(G\), and transition functions \(\operatorname{Ad} p_{ij}\). Since \(dp_{ij}=0\), it follows from (3) that connections in the bundle \(P\) are nothing but 1-forms on \(X\) with values in the vector bundle \(\operatorname{Ad} P\). We denote the space of all such forms by \(\mathscr L^1(\operatorname{Ad} P)\). The condition that the curvature be zero is expressed for \(\alpha\in\mathscr L^1(\operatorname{Ad} P)\) by the formula

\[ d\alpha+\frac{1}{2}[\alpha,\alpha]=0 \]

(the commutation has meaning, since \(\operatorname{Ad} P\) is a bundle of Lie algebras).

Denote the set of all such forms \(\alpha\) again by \(Z(P)\). One can verify that, when connections are identified with forms from \(\mathscr L^1(\operatorname{Ad} P)\), the operators \(L_f^*\) \((f\in D(\operatorname{Int} P))\) act according to the formula

\[ L_f^*\alpha=\operatorname{Ad}(f^{-1})\alpha+f^{-1}df. \]

It is therefore clear that \(L_f^*\) coincides with the operator \(C(f^{-1})\), defined in [1]. By Theorem 3 of that paper, we have a one-to-one mapping \(p:H(P)\to H^1(X,\mathfrak G)\), whose image consists of all locally constant bundles differentiably equivalent to the bundle \(P\). Consider the diagram

\[ \begin{array}{ccc} H(P) & \xrightarrow{\ \bar\mu\ } & \Omega(X,\mathfrak G)\\ {\scriptstyle p}\searrow & & \swarrow{\scriptstyle q}\\ & H^1(X,\mathfrak G) & \end{array} \]

It is verified that it is commutative. Theorem 1 follows immediately from this.

1. Let us consider the case in which the bundle \(P\) is trivial. It follows from Theorem 1 that a homomorphism \(\omega:\pi_1(X)\to\mathfrak G\) is a monodromy homomorphism of an equation of the form (1) if and only if the corresponding bundle \(P_\omega\) under the mapping \(q\) is differentiably trivial. To decide the question of the triviality of the bundle \(P_\omega\), one may use obstruction theory. Suppose that the group \(\mathfrak G\) is connected, and denote by \(\widetilde{\mathfrak G}\) its universal covering group. If \(\mathfrak G\) is not simply connected, then the first obstruction \(k_\omega\), arising in the construction of a section in \(P_\omega\), lies in the group \(H^2(X,\pi_1(\mathfrak G))\). It is known that \(k_\omega=0\) if and only if the group of the bundle \(P_\omega\) can be “lifted” to \(\widetilde{\mathfrak G}\) [5]. In our case this is equivalent to representability of the homomorphism \(\omega\) in the form \(\omega=\pi\widetilde{\omega}\), where \(\pi:\widetilde{\mathfrak G}\to\mathfrak G\) is the covering, and \(\widetilde{\omega}:\pi_1(X)\to\widetilde{\mathfrak G}\) is some homomorphism. Hence the following assertion is obtained.

Theorem 2. Let \(\mathfrak G\) be a connected Lie group. In order that a homomorphism \(\omega:\pi_1(X)\to\mathfrak G\) be a monodromy homomorphism for an equation of the form (1), it is necessary that it be representable in the form \(\omega=\pi\widetilde{\omega}\), where \(\widetilde{\omega}:\pi_1(X)\to\widetilde{\mathfrak G}\) is some homomorphism. In the following particular cases this condition is sufficient:

a) \(\pi_k(\mathfrak G)=0\) \((k>1)\) (i.e. the maximal compact subgroup in \(\mathfrak G\) is abelian);

b) \(H^k(X,\mathbb Z)=0\) \((k\geq 4)\).

We give examples of results following from this theorem.

Corollary 1. If \(\pi_1(X)\) is a free group with a finite or countable number of generators and \(\mathfrak G\) is a connected Lie group, then every homomorphism \(\pi_1(X)\to\mathfrak G\) is a monodromy homomorphism.

Corollary 2. If \(\pi_1(X)\) is a free abelian group with a finite number of generators and \(\mathfrak G=GL(n,\mathbb C)\), \(SL(n,\mathbb C)\), or \(Sp(n,\mathbb C)\), then every homomorphism \(\pi_1(X)\to\mathfrak G\) is a monodromy homomorphism.

On the other hand, for \(\mathfrak G=GL(n,\mathbb R)\) it is easy to give an example of a homomorphism that is not a monodromy homomorphism.

1. Suppose now that \(X\) is a complex manifold, \(\mathfrak G\) a complex Lie group, and \(P\) a holomorphic principal bundle with base \(X\) and group \(\mathfrak G\). If one considers holomorphic infinitesimal connections with zero curvature in the bundle \(P\), then one obtains a picture analogous to that considered in §§ 1 and 2. In particular, if by \(Z_a(P)\) one denotes the set of all holomorphic connections with zero curvature, then the group of all holomorphic sections of the bundle \(\operatorname{Int}P\) acts on \(Z_a(P)\). Let \(H_a(P)\) be the set of orbits of this group in \(Z_a(P)\). Then there arises a mapping

\[ \mu_a:\ H_a(P)\to \Omega(X,\mathfrak G). \]

Similarly to Theorem 1, one proves

Theorem 3. The mapping \(\mu_a\) is one-to-one, and its image consists of all classes of homomorphisms \(\pi_1(X)\to \mathfrak{G}\) for which the corresponding fiber spaces are analytically equivalent to \(P\).

Let the fiber space \(P\) be trivial. Then the image of \(\mu_a\) consists of the classes of homomorphisms of monodromy of equations of the form (1), where \(\alpha\) is a holomorphic 1-form with values in \(G\). In the case \(\dim_{\mathbb C} X=1\), the question considered by us is closely connected with the classical Riemann–Hilbert problem. In this case the assertion of Theorem 3 on the image \(\mu_a\) was proved by Röhrl \((^2)\).

Suppose now that \(X\) is a Stein manifold. Then holomorphic fiber spaces with base \(X\) and group \(\mathfrak{G}\) are analytically equivalent if and only if they are differentiably equivalent \((^6)\). Therefore, in Theorem 3 in this case it is enough to consider homomorphisms that generate fiber spaces differentiably equivalent to \(P\). In particular, we have \(H_a(P)=H(P)\). If \(P\) is trivial, then one may apply the results formulated in § 3.

Moscow State University
named after M. V. Lomonosov

Received
4 VI 1964

REFERENCES

1. A. L. Onishchik, DAN, 141, No. 4, 803 (1961).
2. H. Röhrl, Math. Ann., 133, No. 1 (1957).
3. A. Likhnerovich, Theory of Connections in the Large and Holonomy Groups, Moscow, 1960.
4. N. Steenrod, The Topology of Fibre Bundles, Moscow, 1953.
5. A. Atagapol, C. R., 246, No. 26, 3570 (1958).
6. H. Grauert, Math. Ann., 135, 263 (1958).