We investigate Lie point symmetries of a system of four nonlinear secondorder ordinary differential equations (ODEs), appropriated to the geodesics of a Drinfel'd double Lie group of the affine Lie group of $\mathbb{R}$. The first integrals associated with Lie point symmetries are obtained by utilizing the constructive method due to Wafo Soh and Mahomed [16]. This method deals with integrability when the symmetry vector fields and the operator associated to the system are unconnected. In certain cases we obtain the explicit expressions of the geodesics. We also show that the geodesics are not complete.

In this paper, we investigate a class of screen conformal invariant lightlike hypersurfaces of an indefinite Sasakian manifold. The geometric configuration of such hypersurfaces is established. We prove that its geometry is closely related to the one of leaves of its conformal screen distributions. We also prove that, in any leaf of a conformal screen distribution of an invariant lightlike hypersurface of an indefinite Sasakian space form, the parallelism and semiparallelism notions are equivalent.

The Heisenberg algebra is deformed with the set of parameters $\{q, l,\lambda\}$ to generate a new family of generalized coherent states respecting the Klauder criteria. In this framework, the matrix elements of relevant operators are exactly computed. Then, a proof on the subPoissonian character of the statistics of the main deformed states is provided. This property is used to determine the induced generalized metric.

We review some results about Bonnet pairs in Minkowski space obtained using split quaternions and split complex numbers. We present also an example of Bonnet pairs of minimal immersed timelike tori with umbilical points. Such examples do not exists in the Euclidean space.

On construit sur une variété compacte un feuilletage de Lie nilpotent et non abélien de codimension 4 qui n'admet pas de déformation résoluble non nilpotente.

Using symmetry arguments, we propose a simple derivation of a fundamental solution of the operator $\partial_t \Delta_H$ in which $\Delta_H$ is Kohn-Laplace operator on the Heisenberg group $H^{2n+1}$. Our derivation extends that of Craddock and Lennox [J. Differential Equations 232(2007), 652-674]. Indeed, these authors solved the same problem by employing a symmetry approach in the case $n=1$ . We demonstrate that the case $n=1$ is quite peculiar from a symmetry standpoint and the extension of symmetry arguments to the case $n>1$ requires some intermediate results.

This is a very brief and partial account of Augustin's mathematics and our common interests. In the end is presented some recent work of mine whose birth he encouraged.

Using the Sasakian join construction with homology 3-spheres, we give a countably infinite number of examples of Sasakian manifolds with perfect fundamental group in all odd dimensions $\geq 3$. These have extremal Sasaki metrics with constant scalar curvature. Moreover, we present further examples of both Sasaki-Einstein and Sasaki$\eta$Einstein metrics.

The question of whether a Sasakian metric can admit an additional compatible ($K$)contact structure is addressed. In the complete case if the second structure is also assumed Sasakian, works of Tachibana-Yu and Tanno show that the manifold must be 3-Sasakian or an odd dimensional sphere with constant curvature. Some extensions of this result are obtained, mainly in dimensions 3 and 5.

It is a great pleasure for me to dedicate this paper to Professor Augustin Banyaga in recognition of both his collegiality and his many contributions to symplectic and contact geometry.

In this paper we survey three approaches to computing the homology of a finite dimensional compact smooth closed manifold using a Morse-Bott function and discuss relationships among the three approaches. The first approach is to perturb the function to a Morse function, the second approach is to use moduli spaces of cascades, and the third approach is to use the Morse-Bott multicomplex. With respect to an explicit perturbation (which can be used to derived the Morse-Bott inequalities), the first two approaches yield the same chain complex up to sign. The third approach is fundamentally different. It combines singular cubical chains and Morse chains in the same multicomplex, which provides a way of interpolating between the singular cubical chain complex and the Morse-Smale-Witten chain complex.

The aim of this note is to study the Osserman condition on two families affine connections. As applications, examples of affine Osserman connections which are Ricci flat but not flat are given.

We show that integrable conformal Jacobi fibrations are in onetoone correspondence with sourcesimply connected fibered conformal contact groupoids. We also prove that prequantizable Poisson fibrations give rise to Jacobi fibrations. In addition, sourcesimply connected symplectic groupoids associated to prequantizable and integrable Poisson fibrations are also prequantizable.

This paper is devoted to the socalled twisted cohomology of Koszul-Vinberg algebras. We discuss relationships between the twisted cohomology of Koszul-Vinberg algebras and Chevalley-Eilenberg cohomology of the commutator algebra of these algebras. We also discuss some geometry applications of these relationships. For instance we obtain some homological criteria for hyberbolicity and for completeness of locally flat manifolds. We also discuss some topics which are related to twisted cohomology. In particular, we use some techniques of information geometry to discuss canonical representations of locally flat connections.

We show that $(S^2\times S^2, \omega_0 \oplus \lambda\omega_0)$, with $\lambda > 1$, is an example of symplectic manifold $(X, \omega)$ such that the $\pi_1{\rm Ham}(X \times X, \omega\oplus \omega)$ contains extra elements than those from $\pi_1{\rm Ham}(X, \omega) \times \pi_1{\rm Ham}(X, \omega)$.

We describe the natural identification of $FH_*(X \times X, \triangle; \omega \oplus \omega)$ with $FH_*(X, \omega)$. Under this identification, we show that the extra elements in ${\rm Ham}(X \times X, \omega \oplus \omega)$ found in [3], for $X = (S^2 \times S^2, \omega_0 \oplus \lambda \omega_0)$ for $\lambda > 1$, do not define new invertible elements in $FH_*(X, \omega)$.