Christopher Croke

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On the generation of smooth three-dimensional rigid body motions. Abstract This paper addresses the problem of generating smooth trajectories between an initial and final position and orientation in space. The main idea is to define a functional depending on velocity or its derivatives that measures the smoothness of a trajectory and find trajectories that(More)
The set of rigid body motions forms the Lie group SE(3), the special Euclidean group in three dimensions. In this paper we investigate Riemannian metrics and aane connections on SE(3) that are suited for kinematic analysis and robot trajectory planning. In the rst part of the paper, we study metrics whose geodesics are screw motions. We prove that no(More)
We prove certain optimal systolic inequalities for a closed Riemannian manifold (X, g), depending on a pair of parameters , n and b. Here n is the dimension of X, while b is its first Betti number. The proof of the inequalities involves constructing Abel-Jacobi maps from X to its Jacobi torus T b , which are area-decreasing (on b-dimensional areas), with(More)
We prove the filling area conjecture in the hyperellip-tic case. In particular, we establish the conjecture for all genus 1 fillings of the circle, extending P. Pu's result in genus 0. We translate the problem into a question about closed ovalless real surfaces. The conjecture then results from a combination of two ingredients. On the one hand, we exploit(More)
We construct a counterexample to a conjectured inequality L ≤ 2D, relating the diameter D and the least length L of a nontrivial closed geodesic, for a Riemannian metric on the 2-sphere. The construction relies on Guillemin's theorem concerning the existence of Zoll surfaces integrating an arbitrary infinitesimal odd deformation of the round metric. Thus(More)
We prove that the flat product metric on D n × S 1 is scattering rigid where D n is the unit ball in R n and n ≥ 2. The scattering data (loosely speaking) of a Riemannian manifold with boundary is map S : U + ∂M → U − ∂M from unit vectors V at the boundary that point inward to unit vectors at the boundary that point outwards. The map (where defined) takes V(More)
We consider compact Riemannian manifolds (M, ∂M, g) with boundary ∂M and metric g on which a finite group Γ acts freely. We determine the extent to which certain rigidity properties of (M, ∂M, g) descend to the quotient (M/Γ, ∂/Γ, g). In particular , we show by example that if (M, ∂M, g) is boundary rigid then (M/Γ, ∂/Γ, g) need not be. On the other hand,(More)
We consider Riemannian metrics on the n-sphere for n ≥ 3 such that the distance between any pair of antipodal points is bounded below by 1. We show that the volume can be arbitrarily small. This is in contrast to the 2-dimensional case where Berger has shown that Area ≥ 1/2. In 1977 Berger [B77] considered the set of Riemannian metrics g on the n-sphere(More)
Acknowledgments I owe my deepest gratitude to my advisor, Professor Christopher Croke, for his excellent guidance and patience. I can always receive constructive comments and warm encouragements when talking with him. Without his guidance and persistent help this dissertation would not have been possible. I would like to thank Professor Herman Gluck for(More)
  • Xiaochen Zhou, Jonathan Block, Wolfgang Ziller, Haomin Wen, Lee Kennard, Marco Radeschi They +1 other
  • 2011
Acknowledgments I would like to thank my thesis advisor, Professor Christopher Croke. Without his help, this thesis could not have existed at all. His guidance on various aspects of mathematical research always impresses me, and encourages me to work hard towards my future career. I can never forget his devotion to geometry and to his students. I personally(More)