• Corpus ID: 118128548

Generalised Surfaces in ${\Bbb{R}}^3$

  title={Generalised Surfaces in \$\{\Bbb\{R\}\}^3\$},
  author={Brendan Guilfoyle and Wilhelm Klingenberg},
  journal={arXiv: Differential Geometry},
The correspondence between 2-parameter families of oriented lines in ${\Bbb{R}}^3$ and surfaces in $T{\Bbb{P}}^1$ is studied, and the geometric properties of the lines are related to the complex geometry of the surface. Congruences generated by global sections of $T{\Bbb{P}}^1$ are investigated and a number of theorems are proven that generalise results for closed convex surfaces in ${\Bbb{R}}^3$. 
3 Citations
Oriented Straight Lines and Twistor Correspondence
AbstractThe tangent bundle to the n-dimensional sphere is the space of oriented lines in $$\mathbb{R}^{n+1}$$. We characterise the smooth sections of $$T S^{n}\rightarrow S^{n}$$ which correspond to
Reflection in a Translation Invariant Surface
We prove that the focal set generated by the reflection of a point source off a translation invariant surface consists of two sets: a curve and a surface. The focal curve lies in the plane orthogonal
Reflection of a wave off a surface
Abstract.Recent investigations of the space of oriented lines in $$ \mathbb{R}^{3} $$ are applied to geometric optics. The general formulae for reflection of a wavefront in a surface are derived and


On the space of oriented affine lines in $$ \mathbb{R}^3 $$
Abstract.We introduce a local coordinate description for the correspondence between the space of oriented affine lines in Euclidean $$ \mathbb{R}^3 $$ and the tangent bundle to the 2-sphere.
Characteristic classes of real manifolds immersed in complex manifolds
Let M be a compact, orientable, k-dimensional real differentia- able manifold and N an n-dimensional complex manifold, where k > n. Given an immersion t: M -- N, a point x E M is called an
Computational Line Geometry
Fundamentals.- Models of Line Space.- Linear Complexes.- Approximation in Line Space.- Ruled Surfaces.- Developable Surfaces.- Line Congruences and Line Complexes.- Linear Line Mappings #x2014
Spinor Treatment of Stationary Space‐Times
A generalized SU(2) spinor calculus is established on the ``background space'' V3 of the stationary space‐time. The method of spin coefficients is developed in three dimensions. The stationary field
Monopoles and geodesics
AbstractUsing the holomorphic geometry of the space of straight lines in Euclidean 3-space, it is shown that every static monopole of chargek may be constructed canonically from an algebraic curve by
Contact 3-manifolds twenty years since J. Martinet's work
© Annales de l’institut Fourier, 1992, tous droits réservés. L’accès aux archives de la revue « Annales de l’institut Fourier » (http://annalif.ujf-grenoble.fr/) implique l’accord avec les conditions
An Approach to Gravitational Radiation by a Method of Spin Coefficients
A new approach to general relativity by means of a tetrad or spinor formalism is presented. The essential feature of this approach is the consistent use of certain complex linear combinations of