Sur les courbes biharmoniques dans les variétés de Walker de dimension trois
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Date
2025-12
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Publisher
UB, FS
Abstract
This work explores biharmonic curves within three-dimensional Walker manifolds, a central topic in modern differential geometry. Biharmonic curves represent a generalization of geodesics, appearing as critical curves of higher-order energy. Their study is motivated by potential applications in mathematical physics, particularly in areas such as general relativity.
The thesis begins with an introduction to the fundamental concepts of differentiable manifolds, Levi-Civita connections, and curvature properties. It then establishes a theoretical framework for Walker manifolds, highlighting their metric structure and characteristic connections.
The analysis focuses on biharmonic curves, defined by a zero energy condition for the associated bitension field. We derive the equations characterizing these curves in the context of Walker manifolds, identifying existence conditions for non-geodesic trajectories.
Finally, explicit examples illustrate the results obtained, underscoring the dynamic richness allowed by the degenerate structure of Walker manifolds. This thesis contributes to the understanding of higher-order energy trajectories, paving the way for future generalizations towards more complex types of curves.
Description
Mémoire présenté et défendu publiquement en vue de l’obtention du diplôme de Master en Mathématiques Fondamentales et Appliquées