Rectification of circles and quaternions

On Rectification of Circles and an Extension of Beltrami's Theorem

The goal of this paper is to describe all local diffeomorphisms mapping a family of circles, in an open subset of $\r^3$, into straight lines. This paper contains two main results. The first is a complete description of the rectifiable collection of circles in $\r^3$ passing through one point. It turns out that to be rectifiable all circles need to pass through some other common point. The second main result is a complete description of geometries in $\r^3$ in which all the geodesics are circles. This is a consequence of an extension of Beltrami's theorem by replacing straight lines with circl…

Maps That Take Lines to Circles, in Dimension 4

We list all analytic diffeomorphisms between an open subset of the 4-dimensional projective space and an open subset of the 4-dimensional sphere that take all line segments to arcs of round circles. These are the following: restrictions of the quaternionic Hopf fibrations and projections from a hyperplane to a sphere from some point. We prove this by finding the exact solutions of the corresponding system of partial differential equations.

On the hyperkaehler/quaternion Kaehler correspondence

A hyperkaehler manifold with a circle action fixing just one complex structure admits a natural a hyperholomorphic line bundle. This forms the basis for the construction of a corresponding quaternionic Kaehler manifold in the work of A.Haydys. We construct in this paper the corresponding holomorphic line bundle on twistor space and compute many examples, including monopole and Higgs bundle moduli spaces. We also show that the bundle on twistor space has a natural meromorphic connection which realizes it as the quantum line bundle for the hyperkaehler family of holomorphic symplectic structures…

具有3个不动点的有向流形上的圆周作用

Circle actions on oriented manifolds with 3 fixed points

研究了具有3个不动点的紧致有向流形上的圆周作用，在各向同性子流形可定向的假设下，证明了8维情形下不动点权重与四元数射影空间作用一致，且不存在此类12维流形。

四元数和四维空间中的超极小曲面

Complex Quaternions and Superminimal Surfaces in Four-Space

利用复四元数代数，将共形超极小曲面表示为全纯零曲线因子分解，并证明超极小性等价于广义高斯映射的一个分量恒定。

Shape analysis of framed space curves

In the elastic shape analysis approach to shape matching and object classification, plane curves are represented as points in an infinite-dimensional Riemannian manifold, wherein shape dissimilarity is measured by geodesic distance. A remarkable result of Younes, Michor, Shah and Mumford says that the space of closed planar shapes, endowed with a natural metric, is isometric to an infinite-dimensional Grassmann manifold via the so-called square root transform. This result facilitates efficient shape comparison by virtue of explicit descriptions of Grassmannian geodesics. In this paper, we exte…