The geometry of whips

In this paper we study geometric aspects of the space of arcs parametrized by unit speed in the $L^2$ metric. Physically this corresponds to the motion of a whip, and it also arises in studying shape recognition. The geodesic equation is the nonlinear, nonlocal wave equation $\eta_{tt} = \partial_s(\sigma \eta_s)$, with $\lvert \eta_s\rvert\equiv 1$ and $\sigma$ given by $\sigma_{ss}- \lvert \eta_{ss}\rvert^2 \sigma = -\lvert \eta_{st}\rvert^2$, with boundary conditions $\sigma(t,1)=\sigma(t,-1)=0$ and $\eta(t,0)=0$. We prove that the space of arcs is a submanifold of the space of all curves, that the orthogonal projection exists but is not smooth, and as a consequence we get a Riemannian exponential map that it continuous and even differentiable but not $C^1$. This is related to the fact that the curvature is positive but unbounded above, so that there are conjugate points at arbitrarily short times along any geodesic. We also compare this metric to an $L^2$ metric introduced by Michor and Mumford for shape recognition on the homogeneous space $\text{Imm}(I, \mathbb{R}^2)/\mathcal{D}(I)$ of immersed curves modulo reparametrizations; we show it has some similar properties (such as nonnegative but unbounded curvature and a nonsmooth exponential map), but that the $L^2$ metric on the arc space yields a genuine Riemannian distance.Comment: 24 page

Kaon and Hyperon Form Factors in Kaon Electroproduction on the Nucleon

The electromagnetic form factors of strange mesons and baryons are studied by means of kaon electroproduction on the nucleon. The response functions that are sensitive to the K0, Lambda, Sigma, and KK*\gamma transition form factors are systematically explored. The effects of these form factors on several response functions are discussed.Comment: 10 pages, Latex2e, 12 postscript figures. Invited talk given at the International Conference on Hypernuclear and Strange Particle Physics (HYP97), Brookhaven National Laboratory, USA, October 13-18, 1997. To be published in Nucl. Phys.

Kaon Electromagnetic Production on Nuclei

The formation and excitation of hypernuclei through kaon photoproduction is reviewed. Basic features of the production process are emphasized. The possibility of extracting new information on hypernuclear structure and on the wave function of the bound $\Lambda$ is discussed. New results are presented for the quasifree production process $A(\gamma, K \Lambda)B$. Observables of this reaction are shown to be sensitive to the $\Lambda$-nucleus final state interaction.Comment: 10 pages, 4 figures. Invited talk given at the International Conference on Hypernuclear and Strange Particle Physics (HYP97), Brookhaven National Laboratory, USA, October 13-18, 1997. To be published in Nucl. Phys.

Missing Resonances in Kaon Photoproduction on the Nucleon

New kaon photoproduction data on a proton, gamma + p --> K+ + Lambda, are analyzed using a multipole approach. The background terms are given in terms of gauge invariant, crossing symmetric, Born diagrams with hadronic form factors, while the resonances are parameterized using Breit-Wigner forms. Preliminary results suggest a number of new resonances, as predicted by many quark model studies. A comparison between the extracted multipoles and those obtained from KAON-MAID is presented.Comment: 10 pages, 6 figures, invited talk at the International Symposium on Electrophotoproduction of Strangeness on Nucleons and Nuclei (SENDAI 03), Sendai, Japan, 16-18 Jun 200