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Publication# Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group

Abstract

We study Sobolev-type metrics of fractional order s a parts per thousand yen 0 on the group Diff (c) (M) of compactly supported diffeomorphisms of a manifold M. We show that for the important special case M = S (1), the geodesic distance on Diff (c) (S (1)) vanishes if and only if . For other manifolds, we obtain a partial characterization: the geodesic distance on Diff (c) (M) vanishes for and for , with N being a compact Riemannian manifold. On the other hand, the geodesic distance on Diff (c) (M) is positive for and dim(M) a parts per thousand yen 2, s a parts per thousand yen 1. For , we discuss the geodesic equations for these metrics. For n = 1, we obtain some well-known PDEs of hydrodynamics: Burgers' equation for s = 0, the modified Constantin-Lax-Majda equation for , and the Camassa-Holm equation for s = 1.

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Diffeomorphism

In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are

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In differential geometry, a Riemannian manifold or Riemannian space (M, g), so called after the German mathematician Bernhard Riemann, is a real, smooth manifold M equipp

Equation

In mathematics, an equation is a mathematical formula that expresses the equality of two expressions, by connecting them with the equals sign . The word equation and its cognates in other languages

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The subject of this thesis lies in the intersection of differential geometry and functional analysis, a domain usually called global analysis. A central object in this work is the group Ds(M) of all orientation preserving diffeomorphisms of a compact manifold M with boundary. One of the main properties of this group is that it can be endowed with the structure of an infinite dimensional Hilbert manifold. The study of this object is motivated by the historical results obtained by Arnold [1966] and Ebin and Marsden [1970] in fluid mechanics. In Arnold [1966], it is shown that the motion of an incompressible fluid on a domain M can be formally described by a geodesic in the group Dμ(M) of all volume preserving diffeomorphisms of M, with respect to an L2 Riemannian metric associated to the kinetic energy of the fluid. The functional analytic study of this point of view was used in Ebin and Marsden [1970] in order to show that the Euler equations u̇ + ∇uu = - grad p, div u = 0, u || ∂M are well-posed in Sobolev spaces with sufficient regularity, for any compact manifold M with boundary. We describe below the main results of this thesis. In Chapter 2 we show that the n-dimensional Camassa-Holm equations ṁ + ∇um + ∇uTm + m div u = 0, m = (1 - α2∆)u are well-posed relative to Dirichlet, Navier-slip, or mixed boundary conditions. In order to obtain this result we use the remarkable fact that the associated Lagrangian motion describes a geodesic on a diffeomorphism group with respect to an H1 Riemannian metric. This allows us to use a method inspired from that of Ebin and Marsden [1970] for the Euler equations. The study of the analytic and geometric properties of Ds(M) is deeply related to results concerning the continuity of composition and multiplication in Sobolev spaces. In Chapter 3, we improve these results and obtain a theorem about the continuity of the multiplication and composition of functions below the critical exponent. This result will be used in a crucial way in Chapter 5. We study the classification of the coadjoint orbits of the Sobolev Bott-Virasoro group, for nonzero central charges. The Bott-Virasoro group is the unique nontrivial central extension of the group D(S1) of all diffeomorphisms of the circle. This group and its Lie algebra (the so called Virasoro algebra) appear in a natural way in different areas of physics and mathematics such as string theory or the Korteweg-de Vries equation. The classification of the Bott-Virasoro coadjoint orbits was carried out in Kirillov [1982], Witten [1988] et Balog, Fehér and Palla [1998] and others. In Chapter 4, we consider the completion of the Bott-Virasoro group with respect to a Sobolev topology and obtain a Hilbert manifold called the Sobolev Bott-Virasoro group. We determine the appropriate regularity assumption that allows an extension of the classification of the coadjoint orbits to the case of the Sobolev Bott-Virasoro, and show that the orbits are submanifolds of a Hilbert space. The universal Teichmüller space T (1), is traditionally endowed with a complex Banach manifold structure, a natural Riemannian metric called the Weil-Petersson metric, and a group structure. The last two structures are not compatible with the Banach manifold structure. Indeed, T (1) is not a topological group and the Weil-Petersson metric is only defined on a subset of the tangent bundle of T (1). This incompatibility problem is solved in Takhtajan and Teo [2006] by showing that T (1) can be endowed with a new complex Hilbert manifold structure compatible with the Weil-Petersson metric and making the connected component of the identity into a topological group. By the Beurling-Ahlfors extension theorem, T (1) can be identified with the group QS(S1)fix of all homeomorphisms of the circle fixing three points. In Chapter 5, we study the Hilbert manifold structure induced on QS(S1)fix and the regularity of the elements in the connected component of the identity. With respect to this structure, the group QS(S1)fix is a manifold modeled on the Sobolev space H3/2(S1) which is exactly the critical space for the study of the diffeomorphism group Ds(S1), s > 3/2. Using the strongness of the Weil-Petersson metric, we obtain global existence and uniqueness of the geodesics. This allows us to apply the Euler-Poincaré reduction process and to obtain the spatial representation of the geodesics, called the Euler-Weil-Petersson equation. We end this chapter by showing how these results can be applied in pattern recognition.

Martin Bauer, Martins Bruveris

In this article we study Sobolev metrics of order one on diffeomorphism groups on the real line. We prove that the space equipped with the homogeneous Sobolev metric of order one is a flat space in the sense of Riemannian geometry, as it is isometric to an open subset of a mapping space equipped with the flat -metric. Here denotes the extension of the group of all compactly supported, rapidly decreasing, or -diffeomorphisms, which allows for a shift toward infinity. Surprisingly, on the non-extended group the Levi-Civita connection does not exist. In particular, this result provides an analytic solution formula for the corresponding geodesic equation, the non-periodic Hunter-Saxton (HS) equation. In addition, we show that one can obtain a similar result for the two-component HS equation and discuss the case of the non-homogeneous Sobolev one metric, which is related to the Camassa-Holm equation.

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.