Let (
, ∇,
,
) be a manifold endowed with a flat torsionless connection
and a Riemannian metric
,
and (
the sequence of tangent bundles given by
=
) and
=
. We show that, for any
≥ 1,
carries a ...Hermitian structure (
,
) and a flat torsionless connection
and when
is a Lie group and (∇,
,
) are left invariant there is a Lie group structure on each
such that (
,
,
) are left invariant. It is well-known that (
,
,
) is Kähler if and only if
,
is Hessian, i.e, in each system of affine coordinates (
, . . .,
),
. Having in mind many generalizations of the Kähler condition introduced recently, we give the conditions on (∇,
,
) so that (
,
,
) is balanced, locally conformally balanced, locally conformally Kähler, pluriclosed, Gauduchon, Vaisman or Calabi-Yau with torsion. Moreover, we can control at the level of (∇,
,
) the conditions insuring that some (
,
,
) or all of them satisfy a generalized Kähler condition. For instance, we show that there are some classes of (
, ∇,
,
) such that, for any
≥ 1, (
,
,
) is balanced non-Kähler and Calabi-Yau with torsion. By carefully studying the geometry of (
, ∇,
,
), we develop a powerful machinery to build a large classes of generalized Kähler manifolds.
In this paper, we generalize all the results obtained on para‐Kähler Lie algebras in 3 to para‐Kähler Lie algebroids. In particular, we study exact para‐Kähler Lie algebroids as a generalization of ...exact para‐Kähler Lie algebras. This study leads to a natural generalization of pseudo‐Hessian manifolds, we call them contravariant pseudo‐Hessian manifolds. Contravariant pseudo‐Hessian manifolds have many similarities with Poisson manifolds. We explore these similarities which, among others, leads to a powerful machinery to build examples of non trivial pseudo‐Hessian structures. Namely, we will show that given a finite dimensional commutative and associative algebra (A,.), the orbits of the action Φ of (A,+) on A∗ given by Φ(a,μ)=exp(La∗)(μ) are pseudo‐Hessian manifolds, where La(b)=a.b. We illustrate this result by considering many examples of associative commutative algebras and show that the resulting pseudo‐Hessian manifolds are very interesting.
Let
(
M
,
⟨
,
⟩
TM
)
be a Riemannian manifold. It is well known that the Sasaki metric on
TM
is very rigid, but it has nice properties when restricted to
T
(
r
)
M
=
{
u
∈
T
M
,
|
u
|
=
r
}
. In this ...paper, we consider a general situation where we replace
TM
by a vector bundle
E
⟶
M
endowed with a Euclidean product
⟨
,
⟩
E
and a connection
∇
E
which preserves
⟨
,
⟩
E
. We define the Sasaki metric on
E
and we consider its restriction
h
to
E
(
r
)
=
{
a
∈
E
,
⟨
a
,
a
⟩
E
=
r
2
}
. We study the Riemannian geometry of
(
E
(
r
)
,
h
)
generalizing many results first obtained on
T
(
r
)
M
and establishing new ones. We apply the results obtained in this general setting to the class of Euclidean Atiyah vector bundles introduced by the authors in Boucetta and Essoufi J Geom Phys 140:161–177, 2019). Finally, we prove that any unimodular three dimensional Lie group
G
carries a left invariant Riemannian metric, such that
(
T
(
1
)
G
,
h
)
has a positive scalar curvature.
On Einstein Lorentzian nilpotent Lie groups Boucetta, Mohamed; Tibssirte, Oumaima
Journal of pure and applied algebra,
December 2020, 2020-12-00, Letnik:
224, Številka:
12
Journal Article
Recenzirano
Odprti dostop
In this paper, we study Lorentzian left invariant Einstein metrics on nilpotent Lie groups. We show that if the center of such Lie groups is degenerate then they are Ricci-flat and their Lie algebras ...can be obtained by the double extension process from an abelian Euclidean Lie algebra. We show that all nilpotent Lie groups up to dimension 5 endowed with a Lorentzian Einstein left invariant metric have degenerate center and we use this fact to give a complete classification of these metrics. We show that if g is the Lie algebra of a nilpotent Lie group endowed with a Lorentzian left invariant Einstein metric with non zero scalar curvature then the center Z(g) of g is nondegenerate Euclidean, the derived ideal g,g is nondegenerate Lorentzian and Z(g)⊂g,g. We give the first examples of Ricci-flat Lorentzian nilpotent Lie algebra with nondegenerate center.
We show that any left invariant metric with harmonic curvature on a solvable Lie group is Ricci-parallel. We show the same result for any Lie group of dimension ≤ 6.
A flat pseudo-Euclidean Lie algebra is a real Lie algebra with a non degenerate symmetric bilinear form and a left symmetric product whose the commutator is the Lie bracket and such that the left ...multiplications are skew-symmetric. We show that the center of a flat pseudo-Euclidean nilpotent Lie algebra of signature (2,n−2) must be degenerate and all flat pseudo-Euclidean nilpotent Lie algebras of signature (2,n−2) can be obtained by using the double extension process from flat Lorentzian nilpotent Lie algebras. We show also that the center of a flat pseudo-Euclidean 2-step nilpotent Lie algebra is degenerate and all these Lie algebras are obtained by using a sequence of double extension from an abelian Lie algebra. In particular, we determine all flat pseudo-Euclidean 2-step nilpotent Lie algebras of signature (2,n−2). The paper contains also some examples in low dimension.
We present a case of a papillary tumour at the cerebellopontine angle in a 54-year-old man. He presented with right-sided ear pain associated with dizziness and hearing loss. The radiological ...diagnosis was in favor of acoustic neurinoma. Surgical excision was performed and the diagnosis of the endolymphatic sac tumour was made. Endolymphatic tumour is a low grade adenocarcinoma that originates from the endolymphatic sac. The definitive diagnosis requires a combination of clinical features, radiological finding and pathological correlation.
A 48-year-old man was referred to our unit for assessment of recurring episodes of painful torticollis. Family and past histories were unremarkable. There was no traumatic antecedent. During the ...previous three years he had experienced several episodes of torticollis and painful cervical movements without radiculopathy. His neurological examination was normal, except for a head tilt, decrease range of cervical motion and local tenderness on the right lateral side of the neck. Cervical spine radiographs showed a cervical scoliosis with right unilateral C5-C6 facet joint hypertrophy (A). Cervical computed tomography-scan and magnetic resonance imaging demonstrated a small bone regular tumor in the right C6 articular process and important amyotrophia of the neck musculature on the right side without nerve root or spinal cord compression (B and C). A posterior cervical approach was performed and the mass was completely removed without facet joint sacrifice. At surgery, the tumor appeared well-circumscribed, firm, and calcified with a cartilaginous-cap like appearance (D). Histological features were consistent with benign osteochodroma. The patient was discharged home pain free and referred for physiotherapy care with a good outcome. Osteochondroma is the most common benign tumor of bone (especially long bones), but the spine is rarely involved and usually indicates a hereditary cause such as osteochondromatosis (hereditary multiple exostosis). As seen in our case, this lesion is slow growing, and therefore significant spinal deformation can occur before the symptoms are recognized.
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