Professeur.Aboubacar NIBIRANTIZA

Nom et prénom         :Aboubacar NIBIRANTIZA

Faculté/Institut         : Institut de Pédagogie Appliquée (IPA)

Centre de recherche    : Centre Universitaire de Recherche et de Pedagogie Appliquees aux Sciences(CURPAS) /Laboratoire des Sciences Exactes et Appliquees (LSEA)

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• The research paper investigates the computation of Casimir operators on sl 2-modules associated with linear differential operators and symbols on the real line. The study considers spaces of differential operators and principal symbols related to tensor densities, and derives explicit formulas for the Casimir operators. It extends previously known results about sl 2 -modules and Schur’s Lemma, enriching the understanding of their role in differential geometry and representation theory.
  publier le 2025-11-25 07:20:39

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• In this paper, we give a new presentation of the classification of simple real Jordan algebras. They are classified in four types: type I, type II, type III and type IV and to obtain those types we have to classify the involution classes of Euclidean simple Jordan algebras. We specify which involution is adapted and this one defines the type of simple real Jordan algebra classified.
  publier le 2025-11-25 07:38:47

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• In this paper,we compute the invariant geometry of the statistical manifold of the normal distributions by using the divergence approach. In order to give the isometries, we investigate the component functions of a Killing vector field for the Fisher information metric on the statistical manifold of Gaussian distributions of probabilities. They are harmonic conjugate and then, they are both harmonic functions. Finally, we describe the form of its Killing vector fields.
  publier le 2025-11-25 07:56:35

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• We establish the links between the lightlike geometry and basics invariants of the associated semi-Riemannian geometry on r-lightlike submanifold and semi-Riemannian constructed from a semi-Riemannian ambient. Then we establish some basic inequalities, involving the scalar curvature and shape operator on r-lightlike coisotropic submanifold in semi-Riemannian manifold. Equality cases are also discussed.
  publier le 2025-11-25 08:07:07

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• The research paper investigates the computation of Casimir operators on sl2-modules associated with linear differential operators and symbols on the real line. The study considers spaces of differential operators and principal symbols related to tensor densities, and derives explicit formulas for the Casimir operators. It extends previously known results about sl 2 -modules and Schur’s Lemma, enriching the understanding of their role in differential geometry and representation theory.
  publier le 2026-01-30 10:03:29

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• In this paper, we show that the Lie superalgebra spo ( 2 l + 2| n ) is into the intersection of Lie superalgebra of contact vector fields K ( 2 l + 1| n ) and the Lie superalgebra of projective vector fields pgl ( 2 l + 2 |n ). Explicitly, we use the embedding of a Lie superalgebra constituted of matrices belonging to gl ( 2 l + 2| n ) into Vect ( R 2 l + 1 |n ) . We generalize thus in superdimension 2 l + 1 ? n , the matrix realization described in [7] on S^1/2 . We mention that the intersection spo ( 2 l + 2 |n ) = pgl ( 2 l + 2 n ) ? K ( 2 l + 1 n ) that we prove here, in super case, has been proved on R^2 l + 2 in even case in [4].
  publier le 2026-01-30 10:12:03

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• We consider the space of differential operators D?? acting between ?- and ?-densities defined on S1|2 endowed with its standard contact structure. This contact structure allows one to define a filtration on D?? which is finer than the classical one, obtained by writting a differential operator in terms of the partial derivatives with respect to the different coordinates. The space D?? and the associated graded space of symbols S? (?=???) can be considered as spo(2|2)-modules, where spo(2|2) is the Lie superalgebra of contact projective vector fields on S1|2. We show in this paper that there is a unique isomorphism of spo(2|2)-modules between S? and D?? that preserves the principal symbol (i.e. an spo(2|2)-equivariant quantization) for some values of ? called non-critical values. Moreover, we give an explicit formula for this isomorphism, extending in this way the results of [Mellouli N., SIGMA 5 (2009), 111, 11 pages] which were established for second-order differential operators. The method used here to build the spo(2|2)-equivariant quantization is the same as the one used in [Mathonet P., Radoux F., Lett. Math. Phys. 98 (2011), 311-331] to prove the existence of a pgl(p+1|q)-equivariant quantization on Rp|q.
  publier le 2026-01-30 10:18:35

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• In this paper, we compute the invariant geometry of the statistical manifold of the normal distributions by using the divergence approach. In order to give the isometries, we investigate the component functions of a Killing vector field for the Fisher information metric on the statistical manifold of Gaussian distributions of probabilities. They are harmonic conjugate and then, they are both harmonic functions. Finally, we describe the form of its Killing vector fields.
  publier le 2026-01-30 10:29:51

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• Lightlike warped product manifolds are considered in this paper. The geometry of lightlike submanifolds is difficult to study since the normal vector bundle intersects with the tangent bundle. Due to the degenerate metric, the induced connection is not metric and it follows that the Riemannian curvature tensor is not algebraic. In this situation, some basic techniques of calulus are not useable. In this paper, we consider lightlike warped product as submanifold of semi-Riemannian manifold and establish some remarkable geometric properties from which we establish some conditions on the algebraicity of the induced Riemannian curvature tensor.
  publier le 2026-01-30 10:32:50

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• We establish the links between the lightlike geometry and basics invariants of the associated semi-Riemannian geometry on r -lightlike submanifold and semi-Riemannian constructed from a semi-Riemannian ambient. Then we establish some basic inequalities, involving the scalar curvature and shape operator on r -lightlike coisotropic submanifold in semi-Riemannian manifold. Equality cases are also discussed.
  publier le 2026-01-30 10:41:06

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• In this paper, we consider Osserman conditions on lightlike warped product (sub-)manifolds with respect to the Jacobi Operator. We define the Jacobi operator for lightlike warped product manifold and introduce a study of lightlike warped product Osserman manifolds. For the coisotropic case with totally degenerates first factor, we prove that this class consists of Einstein and locally Osserman lightlike warped product.
  publier le 2026-01-30 10:42:47

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