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This paper is devoted to the complete classification of space curves under affine transformations in the view of Cartans theorem. Spivak has introduced the method but has not found the invariants. Furthermore, for the first time, we propound a necessary and sufficient condition for the invariants. Then, we study the shapes of space curves with constant curvatures in detail and suggest their applications in physics, computer vision and image processing.
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Classification of curves up to affine transformation in a finite dimensional space was studied by some different methods. In this paper, we achieve the exact formulas of affine invariants via the equivalence problem and in the view of Cartans lemma and then, state a necessary and sufficient condition for classification of n--Curves.
We examine the local geometry of affine surfaces which are locally symmetric. There are 6 non-isomorphic local geometries. We realize these examples as Type A, Type B, and Type C geometries using a result of Opozda and classify the relevant geometries up to linear isomorphism. We examine the geodesic structures in this context. Particular attention is paid to the Lorentzian analogue of the hyperbolic plane and to the pseudosphere.
If $mathcal{M}=(M, abla)$ is an affine surface, let $mathcal{Q}(mathcal{M}):=ker(mathcal{H}+frac1{m-1}rho_s)$ be the space of solutions to the quasi-Einstein equation for the crucial eigenvalue. Let $tilde{mathcal{M}}=(M,tilde abla)$ be another affine structure on $M$ which is strongly projectively flat. We show that $mathcal{Q}(mathcal{M})=mathcal{Q}(tilde{mathcal{M}})$ if and only if $ abla=tilde abla$ and that $mathcal{Q}(mathcal{M})$ is linearly equivalent to $mathcal{Q}(tilde{mathcal{M}})$ if and only if $mathcal{M}$ is linearly equivalent to $tilde{mathcal{M}}$. We use these observations to classify the flat Type~$mathcal{A}$ connections up to linear equivalence, to classify the Type~$mathcal{A}$ connections where the Ricci tensor has rank 1 up to linear equivalence, and to study the moduli spaces of Type~$mathcal{A}$ connections where the Ricci tensor is non-degenerate up to affine equivalence.
Twenty five years ago U. Pinkall discovered that the Korteweg-de Vries equation can be realized as an evolution of curves in centoraffine geometry. Since then, a number of authors interpreted various properties of KdV and its generalizations in terms of centoraffine geometry. In particular, the Backlund transformation of the Korteweg-de Vries equation can be viewed as a relation between centroaffine curves. Our paper concerns self-Backlund centroaffine curves. We describe general properties of these curves and provide a detailed description of them in terms of elliptic functions. Our work is a centroaffine counterpart to the study done by F. Wegner of a similar problem in Euclidean geometry, related to Ulams problem of describing the (2-dimensional) bodies that float in equilibrium in all positions and to bicycle kinematics. We also consider a discretization of the problem where curves are replaced by polygons. This is related to discretization of KdV and the cross-ratio dynamics on ideal polygons.
This paper focuses on the study of open curves in a manifold M, and proposes a reparameterization invariant metric on the space of such paths. We use the square root velocity function (SRVF) introduced by Srivastava et al. in [11] to define a reparameterization invariant metric on the space of immersions M = Imm([0,1], M) by pullback of a metric on the tangent bundle TM derived from the Sasaki metric. We observe that such a natural choice of Riemannian metric on TM induces a first-order Sobolev metric on M with an extra term involving the origins, and leads to a distance which takes into account the distance between the origins and the distance between the SRV representations of the curves. The geodesic equations for this metric are given, as well as an idea of how to compute the exponential map for observed trajectories in applications. This provides a generalized theoretical SRV framework for curves lying in a general manifold M .
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