Abstract
This paper gives a study of a two dimensional version of the theory of normal surfaces; namely, a study o normal curves and their relations with respect to geodesic curves. This study results with a nice discrete approximation of geodesics embedded in a triangulated orientable Riemannian surface. Experimental results of the two dimensional case are given as well.
Highlights
Motivated by the topological theory of normal surface we give in this paper a complete study of the relations between geodesic curves and normal curves embedded in a triangulated Riemannian surface
A surface is normal if it intersects the tetrahedra of a triangulation in a fairly simple manner. It was proved by Haken in [1] that every incompressible surface embedded in a triangulated 3-manifold can be continuously deformed to a normal surface with respect to any given triangulation of the manifold
Having the theory of normal surfaces, it is desirable to interpret minimal surfaces in terms of normal surfaces. In this direction Jaco and Rubinstein presented in [2] a pversion, that is based on normal surfaces, of minimal and least area surfaces in a triangulated 3-manifold
Summary
Motivated by the topological theory of normal surface we give in this paper a complete study of the relations between geodesic curves and normal curves embedded in a triangulated Riemannian surface. Normal surface theory is a topological piecewise linear (pfor short) counterpart of the differential geometric theory of minimal surfaces This theory studies the ways surfaces intersect with a given triangulation of a 3-manifold. That pminimal and pleast area surfaces share many properties of classical minimal and least area surfaces These results are the first to make precise the analogy between minimal and normal surfaces. An affirmative answer to the above question gives an alternative topological-combinatorial proof of many classical existence results of least area incompressible surfaces in 3-manifolds, such as those obtained for example in [3], using partial differential equations. Theorem 1 gives rise to a curve shortening flow Such flows are of major importance both theoretically and applicatively speaking, (see [4,5,6,7,8]). In Appendix A we will give a brief introduction to the theory of normal surfaces
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