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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
4-step Carnot spaces and the 2-stein condition
María J. Druetta
Dedicated to the memory of my goddaughter María Pía
Abstract: We consider the 2-stein condition on k-step Carnot spaces S. These spaces are a subclass in the class of solvable Lie groups of Iwasawa type of algebraic rank one and contain the homogeneous Einstein spaces within this class. They are obtained as a semidirect product of a graded nilpotent Lie group N and the abelian group R.
We show that the 2-stein condition is not satisfied on a proper 4-step Carnot spaces S.
2000 Mathematics Subject Classification. 53C30, 53C55
Key words and phrases: 2-stein spaces, k-step Carnot spaces, Einstein spaces, Lie groups of Iwasawa type, Damek-Ricci spaces
A Riemannian manifold is said to be a
-stein space, if there exist functions
defined on
such that
![]() |
Here, denote the Jacobi operator associated to
defined by
for all
where
is the curvature tensor of
Harmonic riemannian manifolds are necessarily
-stein.
A -step Carnot space (
) is a simply connected solvable Lie group
which is a semidirect product of a nilpotent Lie group
and the abelian group
Assume that
and
have associated Lie algebras
and
, respectively.
has the left invariant metric induced by the one given on
, where
is a solvable metric Lie algebra
with inner product
such that:
(i) with
and
(ii) has an orthogonal decomposition
into
subspaces given by
![]() |
for some positive constant
Note that, since the adjoint representation ad is a derivation of
the above decomposition defines a graded Lie algebra structure of
, that is,
![]() |
with the convention for
In particular,
is a (
-step nilpotent Lie algebra, that is, the (
-th derived algebra vanishes:
Note, that up to a constant in the metric, we may suppose that In this terminology, the
-Carnot spaces are the well known Carnot spaces and the case of a
-step Carnot space corresponds to the hyperbolic symmetric space
Moreover, due to a result of [6], any homogeneous Einstein space of Iwsawa type and rank one is a
- step Carnot space for some
We remark that the -stein condition was studied in [1] and [3] on Lie groups
of Iwasawa type in the case of
-step nilpotent
, and in particular, on Carnot spaces: in this class those which are
-stein are exactly the Damek-Ricci spaces (also the harmonic ones). Furthermore, in [7] is shown that a simply connected homogenous harmonic space is a Carnot space, which in turn is equivalent to
be a Damek-Ricci space.
Assume that is a
-step Carnot space. Let
be the Lie algebra of
: that is
is a solvable metric Lie algebra
with
with
a graded nilpotent Lie algebra as described above.
Let denote the center of
and let
be the orthogonal complement of
with respect to the metric
restricted to
Thus
decomposes
and note that ad
hence ad
since ad
is symmetric.
For any the skew-symmetric linear operator
is defined by
![]() |
The Levi Civita connection and the curvature tensor on can be computed by,
for any in
.
Recall that since and
are ad
-invariant, they also have decompositions into eigenspaces as
with
,
and the equality
may be possible
Moreover,
is said to be
-step nilpotent if
Here,
and
denote the eigenvalues of
and
, respectively, that we assume they are natural numbers.
1.1. Properties of the operator
. We recall that for any
so that
and
isomorphically. Consequently, as follows from [5, Lemma1.1])
![]() |
1.2. The curvature formulas. By applying the connection formula, one obtains and if
and
, then
and
in case of
-step nilpotent
. Consequently, by a direct computation we obtain the following formulas involving curvatures (see [2, Section 2]).
(i) ad
(ii) If either or
![]() |
(iii) If for any
and
then
![]() |
(iv) Let denote the maximum eigenvalue of
. If
then
1.3. The Einstein condition. Assume that is an Einstein space, that is Ric
for all
. Let
and
be orthonormal bases of
and
, respectively, with
and
. We use the well known formulas that hold in a solvable metric Lie algebra
,
Note that the first Einstein condition becomes
![]() | (1) |
In general, for a unit vector
the maximun eigenvalue of
, we have
![]() | (2) |
The last one is a direct computation by applying the above expressions of in (iv) to
![]() |
having into account that the following equality holds,
![]() |
In fact, this follows by computing
since for any
1.4. The 2-stein condition. For any , the Jacobi operator
associated to
is the symmetric endomorphism of
defined by
We said that
is a
-stein space, or equivalently
(or
) satisfies the
-stein condition, if there exist
![]() |
In particular, -stein spaces are Einstein: Ric
tr
a constant, for all
, and harmonic riemannian spaces are necessarily
-stein. The
-stein condition was studied on Carnot spaces in [1] and [3]: the
-stein Carnot spaces are exactly the Damek-Ricci spaces (also the harmonic ones) in this class.
The proposition below express the -stein condition in an adequate form for our purposes. Let
denote the set of eigenvalues of
.
Proposition 1.1. Assume that satisfies the
-stein condition. If
is a unit vector, then
![]() |
In particular, if Id
![]() | (3) |
Proof. It follows from the same argument as those used in Theorem 4.1 of [4] for (See also [5, Proposition 1.2]). □
2. The 2-stein condition on 4-step Carnot spaces
We consider the case of a -step Carnot space; that is,
as above with
. Hence, by the propeties of
and
We observe that:
and
Thus,
since
that is
is
-step nilpotent and
. Since
we have that
Moreover, from the facts that
and
are ad
-invariant, it follows that
![]() |
We set
Following with the notation introduced in the previous section we have,
Lemma 2.1. If is a
-step Carnot space that satisfies the
-stein condition, then
is either a Damek-Ricci space or, up to scaling, the symmetric hyperbolic space
or
with
If
the equalities
,
hold in the decomposition of
.
Proof. ()
if the Einstein condition is satisfied.
If
for all
since
and
if
Hence,
and tr
and the Einstein condition (1) implies that
Thus, and consequently
In this case
is abelian and
corresponds to the symmetric hyperbolic space
with
.
The previous fact implies that the eigenvalue is not achieved in
, when
is
-step properly. Thus,
![]() |
()
if the
-stein condition holds.
In fact, assume that there exists . In this case
if
condition (1) gives
Moreover, since and
for all
,
, we have that
and
Now, we apply the formula given in Proposition 1.1 for
and
We first note that,
![]() |
since and
for
This implies that tr(
tr
and consequently, equality holds in the Cauchy-Schwartz Inequality
Therefore, ad
in
with
by the Einstein condition. The equality
, implies that
since
and
Id
Hence,
with
,
and
is a Carnot space. In this case
is a Damek-Ricci space (see [3, Theorem 3.1]).□
Assume that If
satisfies the
-stein condition then
and
The following proposition is basic for our purposes,
Proposition 2.2. If is a
-step Carnot space that is
-stein, then
and
Id for any unit vector
. Equivalently,
for all unit vectors
and
.
Proof. Next we will show that ,
![]() |
for any unit vector
Let be a unit vector. The Einstein condition (1) applied to
gives,
![RZ |𝔳 = - 1j2Z - 3 adH |𝔳, 4](/img/revistas/ruma/v47n1/1a02311x.png)
and applying (4),
Thus, (3) gives
and consequently,
![]() | (5) |
Now, using that tr
tr
(see 1.1), (4) and (5), that is
is obtained, since
![]() |
Finally, we will show the last assertion of the proposition. Using that again
we compute
![]() |
by developing Thus,
and substituting the values of Id, we have
since trtr
Note, that the same argument used to show this equality, implies
![]() | (6) |
Therefore, the -stein condition, tr
tr
gives
By applying (4), the last equality is equivalent to
It follows from (6) that
![]() |
and from the fact
![]() |
The final assertion follows as claimed, since
![]() |
implies that equality holds in the Cauchy-Schwartz Inequality
![]() |
which gives Id
or equivalently,
for all unit vector
□
Let be a
-step Lie algebra so that
and
In what follows let
and
be any orthonormal bases of
,
and
, respectively. The following basic formulas are deduced from the hypothesis
and not assuming that the
-stein condition is satisfied. They are shown in [5, Proposition 2.3].
If and
are unit vectors, then
![]() | (7) |
![]() | (8) |
In order to show that the -stein condition is not satisfied by
, in case that
is
-step properly, we need to compute Ric
and Ric
for unit vectors
and
Lemma 2.3. If is a unit vector then,
![]() |
Proof. Let be a unit vector. We first note, tha it is a direct computation to see that
Therefore, for unit vectors ,
and
using the definition of
we obtain
Now, it is a strighforward computation using the definition of to see that,
![]() |
and
Next, by computing
it is immediate that
By applying the equalities given by (7) and (8) we have,
![]() |
the expression stated in the lemma. □
Corollary 2.4. If satisfies the Einstein condition, then for all unit vectors
and
we have,
Proof. (i) and (ii) are obtained by applying the Einstein condition trtr
ad
(2) to
and tr
tr
ad
to
(Lemma 2.3), respectively. In fact, they are immediate since
and
implying the required formulas. □
Theorem 2.5. If is a
-step Carnot space that satisfies the
-stein condition, then
is either, up to scaling, the hyperbolic space
or a Damek-Ricci space.
Proof. Let be a
-step Carnot space with associated Lie algebra
. If
then either
is, up to scaling, the hyperbolic symmetric space
with
or
according to
or
or
is a Damek-Ricci space by [3, Theorem 3.1].
Assume that we will show that
cannot be
-stein unless
(i) Let a unit vector. If
is the operator on
defined by
it follows that
![]() | (9) |
where span
is the totally geodesic subalgebra of
which corresponds to the symmetric hyperbolic space
of constant curvature
We show this expression (9) following the same argument developed in [5, Proposition 1.2]. In order to do that we need to revise some properties fulfilled by
and
First, note that
![]() |
hence since it is symmetric.
Next, by using the definition of for any unit vectors
,
and
we compute:
Thus, and
This property, togheter with the fact
and
imply that (9) holds, since
![]() |
(ii) Next, (9) it will be applied, computing separately each term and using (7) and (8). First, from the above computations we obtain
Hence,
which implies that
![]() |
It is a straightforward computation to see that,
and (9) becomes
Thus,
![]() | (10) |
(iii) Now, from the equality given by Corollary 2.4 (ii), we obtain
![]() | (11) |
and the expression (10)(11) gives
![]() |
Finally, it follows from (7) that
![]() |
and we get a contradiction, since for any unit vectors
and
In order to show this last assertion, note that
isomorphically (1.1) and by Proposition 2.2, a unit vector
is expressed as
with
(
where
in terms of the orthonormal basis
and
Hence
and we have that
□
[1] Benson P., Payne T., Ratcliff G.; Three-step harmonic solvmanifolds. Geometriae Dedicata 101, 2003, 103-127. [ Links ]
[2] Dotti I, Druetta M.J.; Negatively curved homogeneous Osserman spaces. Differential Geometry and Applications 11, 1999, 163-178. [ Links ]
[3] Druetta M.J.; On harmonic and 2-stein spaces of Iwasawa type. Differential Geometry and its Applications 18, 2003, 351-362. [ Links ]
[4] Druetta M.J.; Carnot spaces and the k-stein condition. Advances in Geometry 6, 2006, 447-473. [ Links ]
[5] Druetta M.J.; The 2-stein condition on generalized Carnot spaces, 2005 (pre-print). [ Links ]
[6] Heber J.; Noncompact homogeneous Einstein spaces. Inventiones Math. 133, 1998, 279-352. [ Links ]
[7] Heber J.; On harmonic and asymptotically harmonic homogeneous spaces. Geometric and Functional Analysis 16 (4), 2006, 869-890. [ Links ]
María J. Druetta
CIEM-Conicet y FAMAF,
Universidad Nacional de Córdoba,
Córdoba 5000, Argentina
druetta@mate.uncor.edu
Recibido: 26 de enero de 2006
Aceptado: 1 de noviembre de 2006