Jeanne N. Clelland

From Frenet to Cartan: The Method of Moving Frames

This book is an introduction to the method of moving frames as developed by Cartan, at a level suitable for beginning graduate students familiar with the geometry of curves and surfaces in Euclidean space. The main focus is on the use of this method to compute local geometric invariants for curves and surfaces in various 3-dimensional homogeneous spaces, including Euclidean, Minkowski, equi-affi ne, and projective spaces. Later chapters include applications to several classical problems in differential geometry, as well as an introduction to the nonhomogeneous case via moving frames on Riemannian manifolds.

Geometry of Control-Affine Systems ⋆

Motivated by control-affine systems in optimal control theory, we introduce the notion of a point-ane distribution on a manifold X - i.e., an affine distribution F together with a distinguished vector field contained in F. We compute local invariants for point-affine distributions of constant type when dim(X) = n, rank(F) = n 1, and when dim(X) = 3, rank(F) = 1. Unlike linear distributions, which are characterized by integer- valued invariants - namely, the rank and growth vector - when dim(X) 4, we find local invariants depending on arbitrary functions even for rank 1 point-affine distributions on manifolds of dimension 2.

Parametric backlund transformations. I: Phenomenology

We begin an exploration of parametric Backlund transformations for hyperbolic Monge-Ampere systems. (The appearance of an arbitrary parameter in the transformation is a feature of several well-known completely integrable PDEs.) We compute invariants for such transformations and explore the behavior of four examples, two of which are new, in terms of their invariants, symmetries, and conservation laws. We prove some preliminary results and indicate directions for further research.

Geometry of Optimal Control for Control-Affine Systems

Motivated by the ubiquity of control-affine systems in optimal control theory, we investigate the geometry of point-affine control systems with metric structures in dimen- sions two and three. We compute local isometric invariants for point-affine distributions of constant type with metric structures for systems with 2 states and 1 control and systems with 3 states and 1 control, and use Pontryagins maximum principle to find geodesic tra- jectories for homogeneous examples. Even in these low dimensions, the behavior of these systems is surprisingly rich and varied.

The geometry of lightlike surfaces in Minkowski space

Abstract We investigate the geometric properties of lightlike surfaces in the Minkowski space R 2 , 1 , using Cartan’s method of moving frames to compute a complete set of local invariants for such surfaces. Using these invariants, we give a complete local classification of lightlike surfaces of constant type in R 2 , 1 and construct new examples of such surfaces.

On the intermediate integral for Monge-Ampère equations

Goursat showed that in the presence of an intermediate integral, the problem of solving a second-order Monge-Ampere equation can be reduced to solving a first-order equation, in the sense that the generic solution of the first-order equation will also be a solution of the original equation. An attempt by Hermann to give a rigorous proof of this fact contains an error; we show that there exists an essentially unique counterexample to Hermanns assertion and state and prove a correct theorem.

Geometry of Centroaffine Surfaces in R 5

We use Cartans method of moving frames to compute a complete set of local invariants for nondegenerate, 2-dimensional centroaffine surfaces in

Geometry of Centroaffine Surfaces in R5

\mathbb{R}^5 \setminus \{0\}

A tale of two arc lengths: Metric notions for curves in surfaces in equiaffine space

with nondegenerate centroaffine metric. We then give a complete classification of all homogeneous centroaffine surfaces in this class.

Geometry of centroaffine surfaces in

We use Cartans method of moving frames to compute a complete set of local invariants for nondegenerate, 2-dimensional centroaffine surfaces in