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179
Riemannian geometries on spaces of plane curves
 J. Eur. Math. Soc. (JEMS
"... Abstract. We study some Riemannian metrics on the space of smooth regular curves in the plane, viewed as the orbit space of maps from S1 to the plane modulo the group of diffeomorphisms of S1, acting as reparameterizations. In particular we investigate the metric for a constant A> 0: ..."
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Cited by 104 (27 self)
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Abstract. We study some Riemannian metrics on the space of smooth regular curves in the plane, viewed as the orbit space of maps from S1 to the plane modulo the group of diffeomorphisms of S1, acting as reparameterizations. In particular we investigate the metric for a constant A> 0:
An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach
 Applied and Computational Harmonic Analysis, 2007. doi: 10.1016/j.acha.2006.07.004. URL http://www.mat.univie.ac.at/~michor/curveshamiltonian.pdf
"... Abstract. Here shape space is either the manifold of simple closed smooth unparameterized curves in R 2 or is the orbifold of immersions from S 1 to R 2 modulo the group of diffeomorphisms of S 1. We investige several Riemannian metrics on shape space: L 2metrics weighted by expressions in length a ..."
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Cited by 44 (20 self)
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Abstract. Here shape space is either the manifold of simple closed smooth unparameterized curves in R 2 or is the orbifold of immersions from S 1 to R 2 modulo the group of diffeomorphisms of S 1. We investige several Riemannian metrics on shape space: L 2metrics weighted by expressions in length and curvature. These include a scale invariant metric and a Wasserstein type metric which is sandwiched between two lengthweighted metrics. Sobolev metrics of order n on curves are described. Here the horizontal projection of a tangent field is given by a pseudodifferential operator. Finally the metric induced from the Sobolev metric on the group of diffeomorphisms on R 2 is treated. Although the quotient metrics are all given by pseudodifferential operators, their inverses are given by convolution with smooth kernels. We are able to prove local existence and uniqueness of solution to the geodesic equation for both kinds of Sobolev metrics. We are interested in all conserved quantities, so the paper starts with the Hamiltonian setting and computes conserved momenta and geodesics in general on the space of immersions. For each metric we compute the geodesic equation on shape space. In the end we sketch in some examples the differences between these metrics.
The Differential LambdaCalculus
 Theoretical Computer Science
, 2001
"... We present an extension of the lambdacalculus with differential constructions motivated by a model of linear logic discovered by the first author and presented in [Ehr01]. We state and prove some basic results (confluence, weak normalization in the typed case), and also a theorem relating the usual ..."
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Cited by 44 (9 self)
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We present an extension of the lambdacalculus with differential constructions motivated by a model of linear logic discovered by the first author and presented in [Ehr01]. We state and prove some basic results (confluence, weak normalization in the typed case), and also a theorem relating the usual Taylor series of analysis to the linear head reduction of lambdacalculus.
Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms
 Documenta Math
"... Abstract. The L 2metric or FubiniStudy metric on the nonlinear Grassmannian of all submanifolds of type M in a Riemannian manifold (N, g) induces geodesic distance 0. We discuss another metric which involves the mean curvature and shows that its geodesic distance is a good topological metric. The ..."
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Cited by 39 (23 self)
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Abstract. The L 2metric or FubiniStudy metric on the nonlinear Grassmannian of all submanifolds of type M in a Riemannian manifold (N, g) induces geodesic distance 0. We discuss another metric which involves the mean curvature and shows that its geodesic distance is a good topological metric. The vanishing phenomenon for the geodesic distance holds also for all diffeomorphism groups for the L 2metric. 1.
A metric on shape spaces with explicit geodesics
, 2007
"... Abstract. This paper studies a specific metric on plane curves that has the property of being isometric to classical manifold (sphere, complex projective, Stiefel, Grassmann) modulo change of parametrization, each of these classical manifolds being associated to specific qualifications of the space ..."
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Cited by 31 (14 self)
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Abstract. This paper studies a specific metric on plane curves that has the property of being isometric to classical manifold (sphere, complex projective, Stiefel, Grassmann) modulo change of parametrization, each of these classical manifolds being associated to specific qualifications of the space of curves (closedopen, modulo rotation etc...) Using these isometries, we are able to explicitely describe the geodesics, first in the parametric case, then by modding out the paremetrization and considering horizontal vectors. We also compute the sectional curvature for these spaces, and show, in particular, that the space of closed curves modulo rotation and change of parameter has positive curvature. Experimental results that explicitly compute minimizing geodesics between two closed curves are finally provided
On Köthe sequence spaces and linear logic
 Mathematical Structures in Computer Science
, 2001
"... We present a category of locally convex topological vector spaces which is a model of propositional classical linear logic, based on the standard concept of Kothe sequence spaces. In this setting, the spaces interpreting the exponential have a quite simple structure of commutative Hopf algebra. The ..."
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Cited by 30 (9 self)
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We present a category of locally convex topological vector spaces which is a model of propositional classical linear logic, based on the standard concept of Kothe sequence spaces. In this setting, the spaces interpreting the exponential have a quite simple structure of commutative Hopf algebra. The coKleisli category of this linear category is a cartesian closed category of entire mappings. This work provides a simple setting where typed calculus and dierential calculus can be combined; we give a few examples of computations. 1
Central extensions of infinitedimensional Lie groups, Annales de l’Inst. Fourier 52:5
, 2002
"... Abstract. In the present paper we study abelian extensions of connected Lie groups G modeled on locally convex spaces by smooth Gmodules A. We parametrize the extension classes by a suitable cohomology group H 2 s (G,A) defined by locally smooth cochains and construct an exact sequence that describ ..."
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Cited by 23 (3 self)
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Abstract. In the present paper we study abelian extensions of connected Lie groups G modeled on locally convex spaces by smooth Gmodules A. We parametrize the extension classes by a suitable cohomology group H 2 s (G,A) defined by locally smooth cochains and construct an exact sequence that describes the difference between H 2 s (G,A) and the corresponding continuous Lie algebra cohomology space H 2 c (g,a). The obstructions for the integrability of a Lie algebra extensions to a Lie group extension are described in terms of period and flux homomorphisms. We also characterize the extensions with global smooth sections resp. those given by global smooth cocycles. Finally we apply the general theory to extensions of several types of diffeomorphism groups.
Choosing roots of polynomials smoothly
 Israel J. Math
, 1998
"... Abstract. We clarify the question whether for a smooth curve of polynomials one can choose the roots smoothly and related questions. Applications to perturbation theory of operators are given. Table of contents 1. Introduction........................... ..."
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Cited by 22 (12 self)
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Abstract. We clarify the question whether for a smooth curve of polynomials one can choose the roots smoothly and related questions. Applications to perturbation theory of operators are given. Table of contents 1. Introduction...........................
Geometry of the VirasoroBott group
 J. Lie Theory
, 1998
"... Abstract. We consider a natural Riemannian metric on the infinite dimensional manifold of all embeddings from a manifold into a Riemannian manifold, and derive its geodesic equation in the case Emb(R, R) which turns out to be Burgers ’ equation. Then we derive the geodesic equation, the curvature, a ..."
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Cited by 21 (10 self)
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Abstract. We consider a natural Riemannian metric on the infinite dimensional manifold of all embeddings from a manifold into a Riemannian manifold, and derive its geodesic equation in the case Emb(R, R) which turns out to be Burgers ’ equation. Then we derive the geodesic equation, the curvature, and the Jacobi equation of a right invariant Riemannian metric on an infinite dimensional Lie group, which we apply to Diff(R), Diff(S 1), and the VirasoroBott group. Many of these results are well known, the emphasis is on conciseness and clarity. Table of contents 1. Introduction..........................