Revista de la Unión Matemática Argentina
versión On-line ISSN 1669-9637
Rev. Unión Mat. Argent. v.47 n.1 Bahía Blanca ene./jun. 2006
Paolo Piccione and Daniel V. Tausk
Abstract: Symmetric connections that are compatible with semi-Riemannian metrics can be characterized using an existence result for an integral leaf of a (possibly non integrable) distribution. In this paper we give necessary and sufficient conditions for a left-invariant connection on a Lie group to be the Levi-Civita connection of some semi-Riemannian metric on the group. As a special case, we will consider constant connections in .
2000 Mathematics Subject Classification. 53B05, 53C05
In this short note we address the following problem: given a (symmetric) connection on a smooth manifold , under which conditions there exists a semi-Riemannian metric in which is -parallel? This problem can be studied using holonomy theory (see ). Alternatively, the problem can be cast in the language of distributions and integral submanifolds, as follows. A connection on a manifold induces naturally a connection in all tensor bundles over (see for instance [4, § 2.7]), in particular, on the bundle of all (symmetric) (2,0)-tensors on , say, . If is a -tensor on , and , then the curvature is the bilinear form on given by:
where and is the curvature tensor of . A semi-Riemannian metric is a (globally defined) symmetric nondegenerate -tensor on , and compatibility with is equivalent to the property that the section is everywhere tangent to the horizontal distribution determined by the connection . However, such distribution is in general non integrable, namely, integrability of the horizontal distribution is equivalent to the vanishing of the curvature tensor , which is equivalent to the vanishing of . Hence, the classical Frobenius theorem cannot be employed in this situation. Nevertheless, the existence of simply one integral submanifold of a distribution, or, equivalently, of a parallel section of a vector bundle endowed with a connection, may occur even in the case of non integrable distributions. From (1), one sees immediately that if is a -tensor on , then the condition that vanishes along is equivalent to the condition of anti-symmetry of , for all and all .
Let us consider the case that an open neighborhood of a point of a manifold is ruled by a family of curves issuing from , parameterized by points of some manifold . What this means is that it is given a smooth function , defined on an open subset of , with for all , and that admits a smooth right inverse . Assume that it is given a nondegenerate symmetric bilinear form ; one obtains a semi-Riemannian metric on by spreading with parallel transport along the curves . If the tensor obtained in this way is such that is an antisymmetric bilinear form on for all and all , then is -parallel. The precise statement of this fact is the following:
Proposition 1.1. Let be a smooth manifold, be a symmetric connection on , and be a nondegenerate symmetric bilinear form on . Let be a -parametric family of curves on with a local right inverse ; assume that , for all . For each , we denote by the parallel transport along . Assume that for all the linear operator:
is anti-symmetric with respect to , for all , where
denotes the linear operator corresponding to the curvature tensor of . Then is the Levi-Civita connection of the semi-Riemannian metric on defined by setting:
for all .
Proof. See  □
In the real analytic case, we have the following global result:
Proposition 1.2. Let be a simply-connected real-analytic manifold and let be a real-analytic symmetric connection on . If there exists a semi-Riemannian metric on a nonempty open connected subset of having as its Levi-Civita connection then extends to a globally defined semi-Riemannian metric on having as its Levi-Civita connection.□
The two results above will be used in Sections 3, 4 and 5 to characterize symmetric connections in Lie groups that are constant in left invariant referentials. The case of (Lemma 3.1 and Proposition 3.2), and more specifically the -dimensional case (Proposition 4.9), will be studied with some more detail.
It is an interesting problem to study conditions for the existence, uniqueness, multiplicity, etc., of (symmetric) connections that are compatible with arbitrarily given tensors. It is well known that semi-Riemannian metrics admit exactly one symmetric and compatible connection, called the Levi-Civita connection of the metric. Uniqueness can be deduced also by a curious combinatorial argument, see Corollary 2.2. The next interesting case is that of symplectic forms, in which case one has existence, but not uniqueness. We will start the paper with a short section containing a couple of simple results concerning compatible connections. First, we will show the combinatorial argument that shows the uniqueness of the Levi-Civita connection of a semi-Riemannian metric tensor (Corollary 2.2). Second, we will prove that the existence of a symmetric connection compatible with a nondegenerate two-form is equivalent to the fact that is closed, in which case there are infinitely many symmetric connections compatible with (Lemma 2.3).
Let be a smooth manifold and let be any tensor in ; we will be mostly interested in the case when is a semi-Riemannian metric tensor on (i.e., is a nondegenerate symmetric -tensor), or when is a symplectic form on (i.e., is a nondegenerate closed -form). If is a connection in , i.e., a connection on the tangent bundle , then we have naturally induced connections on all tensor bundles on , all of which will be denoted by the same symbol .
The torsion of is the anti-symmetric tensor
where denotes the Lie brackets of the vector fields and ; is called symmetric if . The connection is said to be compatible with if is -parallel, i.e., when .
Establishing whether a given tensor admits compatible connections is a local problem. Namely, one can use partition of unity to extend locally defined connections and observe that a convex combination of compatible connections is a compatible connection. In local coordinates, finding a connection compatible with a given tensor reduces to determining the existence of solutions for a non homogeneous linear system for the Christoffel symbols of the connection.
It is well known that semi-Riemannian metric tensors admit a unique compatible symmetric connection, called the Levi-Civita connection of the metric tensor, which can be given explicitly by Koszul formula (see for instance ). Uniqueness of the Levi-Civita connection can be obtained by a curious combinatorial argument, as follows.
Suppose that and are connections on ; their difference is a tensor, that will be denoted by :
where and are smooth vector fields on . If both and are symmetric connections, then is symmetric:
Proof. Let be fixed. We have:
Proof. Assume that is a semi-Riemannian metric on , and let and two symmetric connections such that ; for all consider the map given by:
where is the difference . Since is symmetric, then is symmetric in the first two variables. On the other hand, is anti-symmetric in the last two variables:
By Lemma 2.1, , hence , and thus . □
For symplectic forms, the situation changes radically. Among all nondegenerate two-forms, the existence of a symmetric compatible connection characterizes the symplectic ones:
Lemma 2.3. Let be a nondegenerate -form on a (necessarily even dimensional) manifold . There exists a symmetric connection in compatible with if and only if is closed. In this case, there are infinitely many symmetric connections that are compatible with .
Proof. If is closed, i.e., if is a symplectic form on , Darboux theorem tells us that one can find coordinates around every point of such that , which means that is constant in such coordinate system. The (locally defined) symmetric connection which has vanishing Christoffel symbols in such coordinates is clearly compatible with . As observed above, using partitions of unity one can find a globally defined symmetric connection compatible with .
Conversely, if is any symmetric connection in , then is given by , where denotes the alternator; in particular, if there exists a compatible symmetric connection it must be . □
Let be a symmetric bilinear map and consider the symmetric connection on defined by:
for any smooth vector fields , on . We now apply the result of Proposition 1.1 to determine when is the Levi-Civita connection of a semi-Riemannian metric on . Given then the parallel transport along the curve is given by:
where we identify with the linear map . For any , the curvature tensor of is given by:
for all . Applying Proposition 1.1 to the -parametric family of curves with right inverse we obtain the following:
Lemma 3.1. Let be a nondegenerate symmetric bilinear form on . Then extends to a semi-Riemannian metric on having (3) as its Levi-Civita connection if and only if the linear operator:
is anti-symmetric with respect to , for all .□
Given a nondegenerate symmetric bilinear form on we denote by the Lie algebra of all -anti-symmetric endomorphisms of . Given a linear endomorphism of we write:
for all .
Proposition 3.2. Let be a symmetric bilinear map and let be the range of the linear map . A nondegenerate symmetric bilinear form on extends to a semi-Riemannian metric on having (3) as its Levi-Civita connection if and only if:
for all and all .
for all and all . The conclusion follows by observing that:
Corollary 3.3. Let be a symmetric bilinear map and let be the range of the linear map . Denote by the Lie algebra spanned by and by the commutator subalgebra of . If is contained in for some nondegenerate symmetric bilinear form on then extends to a semi-Riemannian metric on having (3) as its Levi-Civita connection.□
If , the Lie algebra is one-dimensional. This observation allows us to show that, for , the condition in the statement of Corollary 3.3 is also necessary for to extend to a semi-Riemannian metric on having (3) as its Levi-Civita connection.
Lemma 4.1. Let be a symmetric bilinear map and let be the range of the linear map . Denote by the Lie algebra spanned by . Then a nondegenerate symmetric bilinear form on extends to a semi-Riemannian metric on having (3) as its Levi-Civita connection if and only if .
Proof. Define a sequence of subspaces of inductively by setting and by taking to be the linear span of all commutators , with , . Using the Jacobi identity it is easy to show that and therefore:
By Proposition 3.2, if extends to a semi-Riemannian metric on having (3) as its Levi-Civita connection then and are contained in . Since is one dimensional, we have either or ; in the first case, for all and in the latter case for all . In any case, and the conclusion follows. □
Proof. Assume that and . Write , with and . It is easy to see that is represented by the matrix in some basis of . We define by setting:
Conversely, if for some then we can choose a basis of such that (4) holds and the matrix of on such basis is of the form . □
Corollary 4.3. Let be a symmetric bilinear map and let be the range of the linear map . Denote by the Lie algebra spanned by . There exists a semi-Riemannian metric on having (3) as its Levi-Civita connection if and only if either or is one-dimensional and it is spanned by an invertible matrix.
Proof. Let denote a generator of , so that , for all , where is an antisymmetric bilinear form on ; clearly, the kernel of is the center of . Since is three-dimensional, the kernel of is either or it is one-dimensional; the first possibility does not occur, since is nonzero. □
Proof. Choose a basis of with in . If then , otherwise ; thus, we can replace with a scalar multiple of so that and relations (1) hold. If , we may assume that and ; again, replacing with a scalar multiple of gives and relations (2) hold. □
In what follows we denote by the Lie algebra of linear endomorphisms of .
Proof. We show that is in . Assume not. By Lemma 4.4, there exists a nonzero element in . Then commutes with and with , which implies that is in the center of ; thus is a nonzero multiple of , contradicting our assumption.
Now implies that spans ; thus possibility (1) in the statement of Corollary 4.5 does not occur for it would imply that is a nonzero multiple of the identity. Hence possibility (2) occurs and we can assume that . □
Proof. Let be a nonzero element in ; clearly, is one-dimensional. We can choose , , with . □
Proof. If were invertible then would imply . A contradiction is obtained by taking traces on both sides. □
Proposition 4.9. Let be a symmetric bilinear map and let be the range of the linear map . Then there exists a semi-Riemannian metric on having (3) as its Levi-Civita connection if and only if , for all . In this case, a semi-Riemannian metric on having (3) as its Levi-Civita connection can be chosen with an arbitrary value at the origin.
Proof. If for all then, by Lemma 4.1, any nondegenerate symmetric bilinear form on extends to semi-Riemannian metric on having (3) as its Levi-Civita connection. Now assume that there exists a semi-Riemannian metric on having (3) as its Levi-Civita connection and denote by the Lie algebra spanned by . By Corollary 4.3, either or is one-dimensional and it is spanned by an invertible matrix. Let us show that the second possibility cannot occur. If then . If then and is three-dimensional, which is not possible. If either or then by Lemmas 4.6 and 4.7 there exist with and such that spans . By Lemma 4.8, is not invertible and we obtain a contradiction. □
Let be a Lie group and be a left-invariant connection on . The connection is determined by a bilinear map , i.e.:
for any left-invariant vector fields , on .
The torsion of is given by:
Observe that is torsion-free if and only if there exists a symmetric bilinear map with , for all . If we identify with the linear map then is torsion-free if and only if is a Lie algebra homomorphism. The curvature tensor of is given by:
observe that the first bracket is the commutator in and the second is the Lie algebra product of .
Given a curve on , we identify vector fields along with curves on by left translation. Using this identification, the parallel transport of along a one-parameter subgroup is given by .
Proposition 5.1. Assume that is torsion-free and let be a nondegenerate symmetric bilinear form. The following condition is necessary and sufficient for the existence of an extension of to a semi-Riemannian metric on a neighborhood of the identity of whose Levi-Civita connection is :
In (5) we have denoted by the Lie subalgebra of consisting of -anti-symmetric linear operators.
Proof. Set and consider the one-parameter family of curves defined by . If is an open neighborhood of the origin of that is mapped diffeomorphically by onto an open neighborhood of the identity of then a local right inverse for can be defined by setting , for all . The conclusion follows from Proposition 1.1. □
Corollary 5.2. Assume that is torsion-free and let be a nondegenerate symmetric bilinear form. If is (connected and) simply-connected then condition (5) is necessary and sufficient for the existence of a globally defined semi-Riemannian metric on whose Levi-Civita connection is .
Lemma 5.3. Condition (5) is equivalent to:
where , for all .
Proof. Replace by in (5) and compute the Taylor expansion in powers of of the corresponding expression. □
 M. P. do Carmo, Riemannian Geometry, Birkhäuser, Boston, 1992. [ Links ]
 S. Kobayashi and K. Nomizu, Foundations of differential geometry. Vol I, Interscience Publishers, New York-London, 1963. [ Links ]
 P. Piccione, D. V. Tausk, The single-leaf Frobenius theorem with applications, preprint 2005, to appear in Resenhas do Instituto de Matemática e Estatística da Universidade de São Paulo. [ Links ]
 P. Piccione, D. V. Tausk, The theory of connections and G-structures: applications to affine and isometric immersions, XV Escola de Geometria Diferencial, Publicações do IMPA, Rio de Janeiro, 2006, ISBN 85-244-0248-2. [ Links ]
Recibido: 27 de septiembre de 2005
Aceptado: 29 de agosto de 2006