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Revista de la Unión Matemática Argentina
versión Online ISSN 16699637
Rev. Unión Mat. Argent. v.47 n.1 Bahía Blanca ene./jun. 2006
Connections compatible with tensors. A characterization of leftinvariant LeviCivita connections in Lie groups
Paolo Piccione and Daniel V. Tausk
Abstract: Symmetric connections that are compatible with semiRiemannian 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 leftinvariant connection on a Lie group to be the LeviCivita connection of some semiRiemannian 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 semiRiemannian metric in which is parallel? This problem can be studied using holonomy theory (see [2]). 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:
 (1) 
where and is the curvature tensor of . A semiRiemannian 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 antisymmetry 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 semiRiemannian 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:
 (2) 
is antisymmetric with respect to , for all , where
denotes the linear operator corresponding to the curvature tensor of . Then is the LeviCivita connection of the semiRiemannian metric on defined by setting:
for all .
Proof. See [3] □
In the real analytic case, we have the following global result:
Proposition 1.2. Let be a simplyconnected realanalytic manifold and let be a realanalytic symmetric connection on . If there exists a semiRiemannian metric on a nonempty open connected subset of having as its LeviCivita connection then extends to a globally defined semiRiemannian metric on having as its LeviCivita 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 semiRiemannian metrics admit exactly one symmetric and compatible connection, called the LeviCivita 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 LeviCivita connection of a semiRiemannian metric tensor (Corollary 2.2). Second, we will prove that the existence of a symmetric connection compatible with a nondegenerate twoform is equivalent to the fact that is closed, in which case there are infinitely many symmetric connections compatible with (Lemma 2.3).
2. Connections compatible with tensors
Let be a smooth manifold and let be any tensor in ; we will be mostly interested in the case when is a semiRiemannian 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 antisymmetric 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 semiRiemannian metric tensors admit a unique compatible symmetric connection, called the LeviCivita connection of the metric tensor, which can be given explicitly by Koszul formula (see for instance [1]). Uniqueness of the LeviCivita 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:
Lemma 2.1. Let be a set and be a map that is symmetric in its first two variables and antisymmetric in its last two variables. Then is identically zero.
Proof. Let be fixed. We have:
Corollary 2.2. There exists at most one symmetric connection which is compatible with a semiRiemannian metric.
Proof. Assume that is a semiRiemannian 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 antisymmetric in the last two variables:

By Lemma 2.1, , hence , and thus . □
For symplectic forms, the situation changes radically. Among all nondegenerate twoforms, 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:
 (3) 
for any smooth vector fields , on . We now apply the result of Proposition 1.1 to determine when is the LeviCivita connection of a semiRiemannian 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 semiRiemannian metric on having (3) as its LeviCivita connection if and only if the linear operator:
is antisymmetric with respect to , for all .□
Given a nondegenerate symmetric bilinear form on we denote by the Lie algebra of all antisymmetric 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 semiRiemannian metric on having (3) as its LeviCivita connection if and only if:
for all and all .
Proof. By Lemma 3.1, extends to a semiRiemannian metric on having (3) as its LeviCivita connection if and only if:
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 semiRiemannian metric on having (3) as its LeviCivita connection.□
If , the Lie algebra is onedimensional. 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 semiRiemannian metric on having (3) as its LeviCivita 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 semiRiemannian metric on having (3) as its LeviCivita 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 semiRiemannian metric on having (3) as its LeviCivita 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. □
Lemma 4.2. Let be a nonzero linear map. There exists a nondegenerate symmetric bilinear form on with if and only if and ; moreover, is positive definite (resp., has index ) if and only if (resp., ).
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:
 (4) 
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 semiRiemannian metric on having (3) as its LeviCivita connection if and only if either or is onedimensional and it is spanned by an invertible matrix.
Proof. Follows from Lemmas 4.1 and 4.2, observing that the elements of have null trace. □
Lemma 4.4. Let be a threedimensional real Lie algebra with onedimensional. Then the center of is onedimensional.
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 threedimensional, the kernel of is either or it is onedimensional; the first possibility does not occur, since is nonzero. □
Corollary 4.5. Let be a threedimensional real Lie algebra with onedimensional. Then there exists a basis of such that one the following commutation relations holds:
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 .
Lemma 4.6. Let be a threedimensional Lie subalgebra of with onedimensional. There exists a basis of with and .
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 . □
Lemma 4.7. If is a twodimensional real Lie algebra with then there exists a basis of with .
Proof. Let be a nonzero element in ; clearly, is onedimensional. We can choose , , with . □
Lemma 4.8. If and then is not invertible.
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 semiRiemannian metric on having (3) as its LeviCivita connection if and only if , for all . In this case, a semiRiemannian metric on having (3) as its LeviCivita 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 semiRiemannian metric on having (3) as its LeviCivita connection. Now assume that there exists a semiRiemannian metric on having (3) as its LeviCivita connection and denote by the Lie algebra spanned by . By Corollary 4.3, either or is onedimensional and it is spanned by an invertible matrix. Let us show that the second possibility cannot occur. If then . If then and is threedimensional, 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. □
5. LeftInvariant Connections on Lie Groups
Let be a Lie group and be a leftinvariant connection on . The connection is determined by a bilinear map , i.e.:
for any leftinvariant vector fields , on .
The torsion of is given by:
Observe that is torsionfree if and only if there exists a symmetric bilinear map with , for all . If we identify with the linear map then is torsionfree 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 oneparameter subgroup is given by .
Proposition 5.1. Assume that is torsionfree and let be a nondegenerate symmetric bilinear form. The following condition is necessary and sufficient for the existence of an extension of to a semiRiemannian metric on a neighborhood of the identity of whose LeviCivita connection is :
 (5) 
In (5) we have denoted by the Lie subalgebra of consisting of antisymmetric linear operators.
Proof. Set and consider the oneparameter 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 torsionfree and let be a nondegenerate symmetric bilinear form. If is (connected and) simplyconnected then condition (5) is necessary and sufficient for the existence of a globally defined semiRiemannian metric on whose LeviCivita connection is .
Proof. It follows from Proposition 5.1 and from Proposition 1.2 observing that leftinvariant objects on a Lie group are always realanalytic. □
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. □
[1] M. P. do Carmo, Riemannian Geometry, Birkhäuser, Boston, 1992. [ Links ]
[2] S. Kobayashi and K. Nomizu, Foundations of differential geometry. Vol I, Interscience Publishers, New YorkLondon, 1963. [ Links ]
[3] P. Piccione, D. V. Tausk, The singleleaf Frobenius theorem with applications, preprint 2005, to appear in Resenhas do Instituto de Matemática e Estatística da Universidade de São Paulo. [ Links ]
[4] P. Piccione, D. V. Tausk, The theory of connections and Gstructures: applications to affine and isometric immersions, XV Escola de Geometria Diferencial, Publicações do IMPA, Rio de Janeiro, 2006, ISBN 8524402482. [ Links ]
Paolo Piccione
Departamento de Matemática,
Universidade de São Paulo,
São Paulo, Brazil.
piccione@ime.usp.br
http://www.ime.usp.br/˜piccione
Daniel V. Tausk
Departamento de Matemática,
Universidade de São Paulo,
São Paulo, Brazil.
tausk@ime.usp.br
http://www.ime.usp.br/˜tausk
Recibido: 27 de septiembre de 2005
Aceptado: 29 de agosto de 2006