Cut locus (Riemannian manifold)

The cleavage site ( English: cut locus ) is a closed subset of a semi - Riemannian manifold and defined relative to another quantity in the manifold. The simplest case is the intersection location of a single point. For manifolds such as the sphere, the torus and the cylinder of the slice location of a point p, q is the set of points in which meet several geodesics that p and q are connected to the same shortest length. More generally, the intersection location of the point p is the closure of the set of intersection points of p. In principle, an intersection point p to q is the exponential of a vector from TpM whose length is the supremum of the interval in which the exponential map is injective. The concept of the cleavage site was first investigated in 1905 by Poincaré.

• 2.1 Examples

Definition

The exact definition of the points of intersection is a function of the distance function of the manifold.

In the Riemannian geometry

In the case of a Riemannian metric of the intersection of the farthest point q along a geodesic is up to the this geodesic is the shortest path from p to q in the entire manifold.

In the Lorentzian geometry

In the Lorentz geometry, a distinction between the Nullschnittort, the time-like cleavage site and the causal (or even non-space -like ) cleavage site. The intersections of q to p in Nullschnittort of p are the points along null geodesics γ of p starting, for which it holds that they are the γ (t ) where the parameter t is the supremum of the interval is, in which the Lorentz distance between p and γ (t) is zero.

For the definition of the future timelike cut place one considers vectors of the tangent bundle TM restricted to the amount of forward time-like unit vectors. This bundle T 1M is also called future - unit bundle. For each of these vectors v from the fiber of the bundle over a point exists one timelike geodesic cv such that their tangent vector v at this point. The range of injectivity of the exponential map can be defined with these notations as follows: A function for which it holds, where d is the Lorentzian distance and π the canonical map from the bundle to the manifold, which is the base point of the vector. The future time-like cleavage site of p is now simply the exponential map to all vectors s (v ) · v, which are based in p and is for s ( v) between 0 and infinity, ie. The causal cleavage site is the union of timelike cut locus with the Nullschnittort.

Properties

The cleavage site contains its definition of the global principle of the distance information about the topology of the manifold. Thus, the section places a point on a topological sphere with Riemannian metric trees and the cleavage sites are on Tori are interlinked rings. In addition, the intersections are closely linked to the principle of conjugate points. So applicable in complete Riemannian manifolds, that a point q of the cleavage site to a point p is either conjugated or there are at least two geodesics with the same shortest length connecting p and q. About this geodesics, there are other sets. When q is not in the scenario described above conjugated to point p at the same time is next to the intersection P in the entire cleavage site of P, then there is a geodesic loop containing both points. If the distance between p and its cleavage site, ie between p and its closest intersection is equal to the Injektivitätsradius the manifold, then this geodesic loop is even a closed geodesic.

Examples

The simplest example of these properties is a cylinder jacket. The geodesics in this manifold are the sections of the cylinder jacket with a plane (ie, circles and arcs ). From a starting point, you can run around in two directions along these arcs around the cylinder. The right and left running around geodesics with the same angle meet after the same length of track along a straight line along the cylinder on the back. This straight line is the slice location. The point of intersection location, which is the starting point of the next, is the one that is right in front of him. These two points are so connected by the theorem at least with a geodesic loop. However, since the cylinders of the Injektivitätsradius is equal to half the circumference, the distance from each point to its cleavage site on the cylinder is equal to the Injektivitätsradius. So there must be a closed geodesic connecting the point and its antipode. This is here plainly satisfied by the circle that passes through both points.

An example where the tree structure in topological spheres is clearly visible, the surface of an abstracted starfish. Cleavage site of the center of the top surface is a star-shaped arrangement of the beams along the arms on the bottom. So the dark lines in the picture. This cleavage site contains the information about the number and length of the arms, wherein each beam is a little shorter than the arm, on which it runs along.