Brylinski, Jean-Luc Radon transform and functionals on the spaces of curves. (English) Zbl 0889.44003 Gelfand, I. M. (ed.) et al., The Gelfand Mathematical Seminars, 1993-1995. Papers from the seminars, held at Rutgers University, New Brunswick, New Jersey, USA and at IHES, Bures-sur-Yvette, France. Boston, MA: Birkhäuser. The Gelfand Mathematical Seminars. 45-73 (1996). The author considers the Radon transform which associates to a 1-form \(\alpha\) on a smooth manifold \(M\) the functional \(I_\alpha\) on the free loop space \(LM\), defined by \(I_\alpha(\gamma)= \int_\gamma\alpha\), where \(\gamma\) is a loop. The main result of the paper is a characterization of the range of the Radon transform as the space of smooth parametrization functionals on \(LM\) satisfying a system of linear partial differential equations (Radon-John system). Finally, a system of nonlinear partial differential equations is derived, which is fulfilled by holonomy functionals. The proof that any solution of this system is a holonomy functional is given only in the abelian case.For the entire collection see [Zbl 0839.00007]. Reviewer: H.-J.Glaeske (Jena) Cited in 1 Document MSC: 44A12 Radon transform 58C06 Set-valued and function-space-valued mappings on manifolds 46F12 Integral transforms in distribution spaces Keywords:inversion formulas; Radon-John system; Radon transform; range; holonomy functionals PDFBibTeX XMLCite \textit{J.-L. Brylinski}, in: The Gelfand Mathematical Seminars, 1993-1995. Papers from the seminars, held at Rutgers University, New Brunswick, New Jersey, USA and at IHES, Bures-sur-Yvette, France. Boston, MA: Birkhäuser. 45--73 (1996; Zbl 0889.44003)