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Publication# Déformations conformes des variétés de Finsler-Ehresmann

Résumé

An intrinsic approach to Finsler geometry is proposed. A concept of Finsler- Ehresmann manifold, denoted by (M,F,H), is introduced and a generalized Chern connection is built for this manifold. Conformal deformations on this manifold are considered. First, we have an analogous of Chern's theorem: we prove the existence and uniqueness of a generalized Chern connection for the manifold (M,F,H). Similarly, within an essentially koszulian formalism, we present two curvatures associated to this generalized connection, namely a R curvature and a P one. The second result is the deduction of conformal transformations laws for the generalized Chern connection and associated curvatures. The transformation of R seems to have very similar properties as that of the Riemannian curvature while that of P reveals other objects of pure Finslerian nature. Third, we construct the finsler Weyl and Schouten tensors W and S respectively and we study their conformal transformations. Furthermore, we show that for the dimension 3, the horizontal component of W for generalized Berwald manifolds is identically zero. The next result is a theorem of Weyl-Schouten type giving necessary and sufficient conditions for a Finsler-Ehresmann manifold to be conformaly R-flat. We complete this result by exploring the case of dimension 3 for Berwald spaces which gives a result very similar to the Riemannian case. In addition, we announce some necessary conditions to characterize conformal flatness of Finsler-Ehresmann manifolds.

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The aim of this dissertation is to solve numerically the following problem, denoted by P : given a Riemannian manifold and two points a and b belonging to that manifold, find a tangent vector T at a, such that expa(T) = b, assuming that T exists. This problem is set under an optimal control formulation, which requires the definition of an objective function and a space of control, the choice of a method for the calculation of the descent direction of that function in the space of control and the use of an optimization algorithm to find its minimum, which corresponds to the solution of the original problem by construction. Several techniques are necessary to be put together, coming from the fields of geometry, numerical analysis and optimization. The first part will concern a recalling of the mathematical context in which this formulation takes place. The general principles of optimal control will also be given. In the second part, we will present an intrinsic formulation of the optimal control problem associated to P, based on Jacobi fields, which will play the role of the so called adjoint state. This derivation leads to necessary optimality conditions. We will illustrate explicitly that formulation by treating the specific case of Riemannian manifolds with constant sectional curvature. Then, we will derive the optimal control problem in coordinates, not only to check the intrinsic formulation but also to reveal how it is hidden behind the expressions in coordinates. Their use reveals some quantities whose interpretation may be given this way. Moreover, we will show that more possibilities exist to chose the cost function and the control space in coordinates. In a second step, an alternative approach will consider the Hamiltonian formulation of geodesics. This is an incursion into symplectic geometry. We will then reformulate the Riemannian optimal control problem in its Hamiltonian version. In the third part, the numerical methods used for solving P will be presented. The discretization imposes the definition of new discrete optimal control problems. The technique shows that the discrete adjoint state equation strongly depends on the numerical scheme used to solve the direct problem. We will give a collection of numerical computations in the specific case of parametric piece of surfaces, where the surface can be defined by one or several Bézier patches, each one corresponding to a chart, which is representative of a Riemannian manifold. We will compare the different numerical approaches. The last but one part will be devoted to the interesting application of wooden roof building, where the structure is made of wooden boards, with geodesic trajectories on the designed piece of surface. The Geos (Geodesic solver) software has been developed for that purpose. After having introduced some specific numerical methods used in the code, we present the Geos application interface (AI) developed as a tool for the conception of such a roof. We then show an existing wooden structure built according to that mean. Finally, we will summarize the results of our research and discuss future possible prospects.

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1992