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A Koszul–Vinberg manifold is a manifold M endowed with a pair (∇ , h) where ∇ is a flat connection and h is a symmetric bivector field satisfying a generalized Codazzi equation. The geometry of such manifolds could be seen as a type of bridge between Poisson geometry and pseudo-Riemannian geometry, as has been highlighted in our previous article [Contravariant Pseudo-Hessian manifolds and their associated Poisson structures. Differential Geometry and its Applications (2020)]. Our objective here will be to pursue our study by focusing in this setting on submanifolds by taking into account some developments in the theory of Poisson submanifolds. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Background: Bone involvement is a common finding in many types of lymphoma. Cranial vault involvement is extremely rare, and the majority of patients are found at staging to have concurrent disease in lymph nodes. Thirty-eight cases of primary lymphoma of the cranial vault have been reported to date. Methods: This article presents a rare case of primary cranial vault lymphoma and conducts a systematic review of the current literature. A total of 36 articles comprising 38 cases were included for analysis. The relevant demographic, clinical, and imaging characteristics, as well as the treatment and outcomes of this unique disease presentation were studied. Results: The average patient age was 60 years. There was no significant difference in patient gender. The predominant patient complaint was a subcutaneous scalp mass. Of the patients, 11.7% were immunocompromised. CT scans showed signs of osteolysis in 74% of lesions and hyperostosis in 5%, and the cranial vault was observed as normal in 18% of cases. MRI was performed in 23 cases. There was a wide range of histological subtypes, with a slight predominance of diffuse large B-cell lymphoma. Treatment consisted of surgery alone, surgery followed by radiotherapy, and surgery followed by radiotherapy and chemotherapy. The follow-up periods ranged from 5 months to 6 years, with a median value of 7 months. Sixteen patients were followed up to 12 months; 13 of them were alive at 1 year from diagnosis. Conclusion: Primary cranial vault lymphoma is an extremely rare finding. It should be considered in the differential diagnosis of scalp masses. Although the analysis of outcome of the reported cases is difficult because of the small number of occurrences of this entity and the variability of follow-up, a combination of surgery, radiotherapy, and chemotherapy seems to offer better outcomes. [ABSTRACT FROM AUTHOR]

We introduce Riemannian Lie algebroids as a generalization of Riemannian manifolds and we show that most of the classical tools and results known in Riemannian geometry can be stated in this setting. We give also some new results on the integrability of Riemannian Lie algebroids. [ABSTRACT FROM AUTHOR]

Eosinophilic granuloma (EG) is a rare benign osteolytic lesion observed rarely in adults, with only some 18 cases of spinal location reported in the literature. We present an unusual variant of EG in a 23-year-old man with radiological features of multiple spinal lytic lesions which was evocated of metastatic processes. A surgically transpedicular biopsy of the thoracic collapsed vertebrae with posterior stabilization was made. Histological examination of the tissue showed features of eosinophilic granuloma. The clinical and radiological findings of EG present dilemmas of both diagnosis and treatment. The etiology is unclear and the therapeutic approach is still controversial. [ABSTRACT FROM AUTHOR]

We show that given a finite-dimensional real Lie algebra $${\mathcal{G}}$$ acting on a smooth manifold P then, for any solution of the classical Yang–Baxter equation on $${\mathcal{G}}$$ , there is a canonical Poisson tensor on P and an associated canonical torsion-free and flat contravariant connection. Moreover, we prove that the metacurvature of this contravariant connection vanishes if the isotropy Lie subalgebras of the action are trivial. Those results permit to get a large class of smooth manifolds satisfying the necessary conditions, introduced by Eli Hawkins, to the existence of noncommutative deformations. [ABSTRACT FROM AUTHOR]

Spinal epidural angiolipoma is a rare benign tumor containing vascular and mature adipose elements. A slow progressive clinical course was mostly presented and rarely a fluctuating course during pregnancy. The authors report the original case of spontaneous spinal epidural bleeding resulting from thoracic epidural angiolipoma who presented with hyperacute onset of paraplegia, simulating an extradural hematoma. The patient was admitted with sudden non-traumatic hyperacute paraplegia during a prolonged walk. Neurologic examination showed sensory loss below T6 and bladder disturbances. Spinal MRI revealed a non-enhanced heterogeneous thoracic epidural lesion, extending from T2 to T3. A bilateral T2–T4 laminectomy was performed to achieve resection of a lipomatous tumor containing area of spontaneous hemorrhage. The postoperative course was uneventful with complete neurologic recovery. Histologic examination revealed the tumor as an angiolipoma. Because the prognosis after rapid surgical management of this lesion is favorable, the diagnosis of spinal angiolipoma with bleeding should be considered in the differential diagnosis of hyperacute spinal cord compression. [ABSTRACT FROM AUTHOR]

The article discusses case study of a 13-year-old girl having complaint of right facial pain. It has been informed that transnasal endoscopy showed edematous nasal mucosa and a firm grayish mass in the right maxillary sinus that extends to sphenoethmoidal sinuses. It has been informed that histology and immunohistochemical staining confirmed Burkitt's lymphoma and she was referred to the oncology department for chemotherapy regimen.

A letter to the editor is presented regarding a case study on the use of fluoxetine to treat postoperative cerebellar mutism in a 13-year-old boy.