In this paper we give a conformal representation of flat surfaces in the hyperbolic 3-space using the complex structure induced by its second fundamental form. We also study some examples and the behaviour at infinity of complete flat ends. Subject Classification (1991): 53A35, 53C42 Mathematics

Our first objective in this paper is to give a natural formulation of the Christoffel problem for hypersurfaces in H n+1 , by means of the hyperbolic Gauss map and the notion of hyperbolic curvature radii for hypersurfaces. Our second objective is to provide an explicit equivalence of this Christoffel problem with the famous problem of prescribing scalar curvature on S n for conformal metrics, posed by Nirenberg and Kazdan-Warner. This construction lets us translate into the hyperbolic setting the known results for the scalar curvature problem, and also provides a hypersurface theory interpretation of such an intrinsic problem from conformal geometry. Our third objective is to place the above result in a more general framework. Specifically, we will show how the problem of prescribing the hyperbolic Gauss map and a given function of the hyperbolic curvature radii in H n+1 is strongly related to some important problems on conformally invariant PDEs in terms of the Schouten tensor. This provides a bridge between the theory of conformal metrics on S n and the theory of hypersurfaces with prescribed hyperbolic Gauss map in H n+1. The fourth objective is to use the above correspondence to prove that for a wide family of Weingarten functionals W(κ 1 ,. .. , κ n), the only compact immersed hypersurfaces in H n+1 on which W is constant are round spheres.

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Abstract-This article reviews several tools we have developed to improve the understanding of locomotor training following spinal cord injury (SCI), with a view toward implementing locomotor training with robotic devices. We have developed (1) a small-scale robotic device that allows testing of locomotor training techniques in rodent models, (2) an instrumentation system that measures the forces and motions used by experienced human therapists as they manually assist leg movement during locomotor training, (3) a powerful, lightweight leg robot that allows investigation of motor adaptation during stepping in response to force-field perturbations, and (4) computational models for locomotor training. Results from the initial use of these tools suggest that an optimal gait-training robot will minimize disruptive sensory input, facilitate appropriate sensory input and gait mechanics, and intelligently grade and time its assistance. Currently, we are developing a pneumatic robot designed to meet these specifications as it assists leg and pelvic motion of people with SCI.

We study isometric immersions of surfaces of constant curvature into the homogeneous spaces H 2 × R and S 2 × R. In particular, we prove that there exists a unique isometric immersion from the standard 2-sphere of constant curvature c > 0 into H 2 ×R and a unique one into S 2 × R when c > 1, up to isometries of the ambient space. Moreover, we show that the hyperbolic plane of constant curvature c < −1 cannot be isometrically immersed into H 2 × R or S 2 × R.MSC: 53C42, 53C40.