Aziz Ikemakhen

• Professor
• Professor (Full) at Cadi Ayyad University

Many applications of geometry modelling and computer graphics necessite accurate curvature estimations of curves on the plane or on manifolds. In this paper, we define the notion of the discrete geodesic curvature of a geodesic polygon on a smooth surface. We show that, when a geodesic polygon P is closely inscribed on a C2\documentclass[12pt]{mini...

We provide a general method for defining and efficiently computing barycentric coordinates with respect to polygons on the unit sphere. More precisely, we develop a novel explicit construction which allows us to compute the spherical barycentric coordinates from their 2D-Euclidean counterparts. In particular, we give two interesting families of sph...

Barycentric coordinates are a fundamental tool in computer graphics and geometry processing. A variety of ways have been proposed for constructing such coordinates on the Euclidean plane. The spherical barycentric coordinates are also developed. This paper completes this construction for the hyperbolic plane case. We define hyperbolic barycentric c...

In recent years, game developers are interested in developing games in the hyperbolic space. Shape blending is one of the fundamental techniques to produce animation and videos games. This paper presents two algorithms for blending between two closed curves in the hyperbolic plane in a manner that guarantees that the intermediate curves are closed....

The study of planar and spherical geometric subdivision schemes was done in Dyn and Hormann (2012); Bellaihou and Ikemakhen (2020). In this paper we complete this study by examining the hyperbolic case. We define general interpolatory geometric subdivision schemes generating curves on the hyperbolic plane by using geodesic polygons and the hyperbol...

Shape morphing is a continuous deformation in time between two shapes (curves, surfaces,..). For planar curves, most efficient methods for blending between two closed curves are based on the construction of the morph curve involving its signed curvature function. The latter is obtained by linear interpolation of the signed curvature functions of th...

Many applications of geometry modeling and computer graphics necessite accurate curvature estimations of curves on the plane or on manifolds. In this paper, we define the notion of the discrete geodesic curvature of a geodesic polygon on a smooth surface. We show that, when a geodesic polygon P is closely inscribed on a $C^2$-regular curve, the dis...

We define general geometric subdivision schemes generating curves on the 2-dimensional unit sphere by us- ing geodesic polygons and the spherical distance. We show that a spherical interpolatory geometric subdi- vision scheme is convergent if the sequence of maximum edge lengths is summable and the limit curve is G 1 -continuous if in addition the...

We give a complete description of semi-symmetric algebraic curvature tensors on a four-dimensional Lorentzian vector space and we use this description to determine all four-dimensional homogeneous semi-symmetric Lorentzian manifolds.

We give a method to lift $(2,0)$-tensors fields on a manifold $M$ to build symplectic forms on $TM$. Conversely, we show that any symplectic form $\Om$ on $TM$ is symplectomorphic, in a neighborhood of the zero section, to a symplectic form built naturally from three $(2,0)$-tensor fields associated to $\Om$.

We provide the tangent bundle TM of pseudo-Riemannian manifold (M, g) with the Sasaki metric gs and the neutral metric gn. First we show that the holonomy group Hs of (TM, gs) contains the one of (M, g). What allows us to show that if (TM, gs) is indecomposable reducible, then the basis manifold (M, g) is also indecomposable-reducible. We determine...

We describe the possible holonomy groups of simply connected Lorentzian spinc manifolds which admit parallel spinors. In particular, we study the case of simply connected Lorentzian symmetric spaces that admit parallel spinors and we give the complete classification of this type of spaces.

We describe, by their holonomy groups, all simply connected irreducible non-locally symmetric pseudo-Riemannian spinc manifolds which admit parallel spinors. So we generalize the Riemannian spinc case [A. Moroianu, Parallel and killing spinors on spinc manifolds, Commun. Math. Phys. 187 (1997) 417–427] and the pseudo-Riemannian spin one [1].

We describe, by their holonomy groups, all simply connected irreducible non-locally symmetric pseudo-Riemannian SpinC manifolds which admit parallel spinors. So we generalise the Riemannian case and the pseudo-Riemannian one.

We describe, by their holonomy groups, all complete simply connected irreducible non-locally symmetric pseudo-Riemannian SpinC manifolds which admit parallel spinors. So we generalize the Riemannian SpinC case and the pseudo-Riemannian Spin one.

We characterize, by their holonomy groups, the totally reducible spin pseudo-Riemannian manifolds which admit parallel spinors. To cite this article: A. Ikemakhen, C. R. Acad. Sci. Paris, Ser. I 339 (2004).

We characterize the spin pseudo-Riemannian manifolds which admit parallel pure spinors by their holonomy groups. In particular, we study the Lorentzian case. To cite this article: A. Ikemakhen, C. R. Acad. Sci. Paris, Ser. I 337 (2003).

In this paper, we determinate a class of possible restricted holonomy groups for a non-irreducible indecomposable pseudo-riemannian manifold with signature $(2,2+n)$. In particular, we deduce that which associated to symmetric spaces; and give some examples of such spaces. Finally, we construct some examples of metrics whose restricted holonomy gro...

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