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Revista de la Unión Matemática Argentina
versión impresa ISSN 0041-6932versión On-line ISSN 1669-9637
Rev. Unión Mat. Argent. v.47 n.1 Bahía Blanca ene./jun. 2006
Connections compatible with tensors. A characterization of left-invariant Levi-Civita connections in Lie groups
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 [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 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:
![]() | (2) |
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 [3] □
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).
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 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 [1]). 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:
Lemma 2.1. Let be a set and
be a map that is symmetric in its first two variables and anti-symmetric in its last two variables. Then
is identically zero.
Proof. Let be fixed. We have:
![ρ](/img/revistas/ruma/v47n1/1a12145x.png)
![ρ](/img/revistas/ruma/v47n1/1a12147x.png)
Corollary 2.2. There exists at most one symmetric connection which is compatible with a semi-Riemannian metric.
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:
![]() | (3) |
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
.
Proof. By Lemma 3.1, extends to a semi-Riemannian metric
on
having (3) as its Levi-Civita 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 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. □
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 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. Follows from Lemmas 4.1 and 4.2, observing that the elements of have null trace. □
Lemma 4.4. Let be a three-dimensional real Lie algebra with
one-dimensional. Then the center
of
is one-dimensional.
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. □
Corollary 4.5. Let be a three-dimensional real Lie algebra with
one-dimensional. 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 three-dimensional Lie subalgebra of
with
one-dimensional. 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 two-dimensional real Lie algebra with
then there exists a basis
of
with
.
Proof. Let be a nonzero element in
; clearly,
is one-dimensional. 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 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. □
5. Left-Invariant Connections on Lie Groups
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
:
![]() | (5) |
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
.
Proof. It follows from Proposition 5.1 and from Proposition 1.2 observing that left-invariant objects on a Lie group are always real-analytic. □
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 York-London, 1963. [ Links ]
[3] 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 ]
[4] 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 ]
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