Cut location

The cleavage site (English: cut locus) is a closed subset of a semi-Riemannian manifold defined and relative to another quantity in the manifold. The simplest case is the intersection of a single point. For manifolds like the sphere , the torus and the cylinder, the intersection of a point is the set of points where several geodesics meet, which connect and with the same shortest length. More generally, the intersection of the point is the completion of the set of intersections of . In principle, an intersection point is to the point the exponential map of a vector from the length of which the supremum is the interval in which the exponential injective is. The concept of the cut location was first examined by Poincaré in 1905 . ${\displaystyle p}$ ${\displaystyle q}$ ${\displaystyle p}$ ${\displaystyle q}$ ${\displaystyle p}$ ${\displaystyle p}$ ${\displaystyle q}$ ${\displaystyle p}$ ${\displaystyle T_{p}M}$

definition

The exact definition of the intersections depends on the distance function of the manifold.

In Riemannian geometry

In the case of a Riemannian metric, the intersection is the most distant point along a geodesic up to which this geodesic represents the shortest connection from to in the entire manifold. ${\displaystyle q}$ ${\displaystyle p}$ ${\displaystyle q}$

In Lorentzian geometry

In Lorentz geometry, a distinction is made between the zero intersection, the time-like intersection and the causal (or non-space-like) intersection. The points of intersection to the Nullschnittort of the points along null geodesics of starting, for which it holds that they are the are in which the parameter is the supremum of the interval in which the distance between Lorentz and zero. ${\displaystyle q}$ ${\displaystyle p}$ ${\displaystyle p}$ ${\displaystyle \gamma }$ ${\displaystyle p}$ ${\displaystyle \gamma \ (t)}$ ${\displaystyle t}$ ${\displaystyle p}$ ${\displaystyle \gamma \ (t)}$

For the definition of the future time-like intersection point, one considers vectors of the tangential bundle restricted to the set of future-oriented time-like unit vectors. This bundle is also called the future unit bundle . For each of these vectors from the fiber of the bundle over a point there is a single time-like geodesic of the kind that its tangential vector is at this point . The range of the injectivity of the exponential mapping can be defined with these notations as follows: A function for which applies , where is the Lorentzian distance and the canonical mapping from the bundle into the manifold that gives the base point of the vector. The future time-like cleavage site of is now simply the exponential to all vectors , which in rest and for lies between 0 and infinity so . The causal intersection is the union of the temporal intersection with the zero intersection. ${\displaystyle TM}$ ${\displaystyle T_{-1}M}$ ${\displaystyle v}$ ${\displaystyle c_{v}}$ ${\displaystyle v}$ ${\displaystyle s\colon T_{-1}M\rightarrow \mathbb {R} \cup \{\infty \}}$ ${\displaystyle s(v)=\operatorname {sup} \{t\geq 0\colon d(\pi (v),c_{v}(t))=t\}}$ ${\displaystyle d}$ ${\displaystyle \pi }$ ${\displaystyle p}$ ${\displaystyle s(v)\cdot v}$ ${\displaystyle p}$ ${\displaystyle s\ (v)}$ ${\displaystyle C_{t}^{+}=\operatorname {exp} _{p}\{s(v)v\colon v\in T_{-1}M|_{p},0<s(v)<\infty \}}$

properties

Intersection location C (P) on the cylinder surface with 2 geodesics of equal length and the manifold that connects P with a point Q in the intersection location. ${\displaystyle \gamma _{1}}$ ${\displaystyle \gamma _{2}}$

The intersection location contains information about the topology of the manifold through its definition via the global principle of distance. The intersection of a point on a topological sphere with Riemannian metrics are trees and the intersection of tori are interconnected rings. In addition, the intersection points are closely related to the principle of conjugate points . In complete Riemannian manifolds, a point of the intersection is either conjugated to a point or there are at least two geodesics with the same shortest length that connect and . There are other sentences about these geodesics. If in the described scenario there is no conjugate point to and at the same time the next intersection point to is in the entire intersection of , then there is a geodetic loop that contains both points. If the distance between and its intersection, i.e. between and its closest intersection, is equal to the injectivity radius of the manifold, then this geodetic loop is even a closed geodesic. ${\displaystyle q}$ ${\displaystyle p}$ ${\displaystyle p}$ ${\displaystyle q}$ ${\displaystyle q}$ ${\displaystyle p}$ ${\displaystyle p}$ ${\displaystyle p}$ ${\displaystyle p}$ ${\displaystyle p}$

Examples

The underside of a starfish, clearly visible are dark grooves along the arms.

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 (i.e. circles and arcs). Starting from one point, one can walk around the cylinder in two directions along these arcs. The right and left circulating geodesics with the same angle meet after the same distance along a straight line along the cylinder on the rear side. This straight line is the cutting point. The point of the cut location that is closest to the starting point is the one that is exactly opposite it. According to the theorem, these two points are connected with at least one geodetic loop. However, since the injectivity radius of the cylinder is equal to half the circumference, the distance from each point to its intersection on the cylinder is equal to the injectivity radius. So there must be a closed geodesic that connects the point and its antipodes. This is obviously fulfilled here by the circle that goes through both points.

An example in which the tree structure in topological spheres can be easily recognized is the surface of an abstract starfish. The intersection to the center of the top is a star-shaped array of rays along the arms on the bottom. So the dark lines in the adjacent picture. This intersection contains the information about the number and length of the arms, whereby each beam is slightly shorter than the arm it is running along.

literature

• J. Beem, P. Ehrlich, K. Easley: Global Lorentzian Geometry , 2nd edition, Marcel Dekker, New York (1996)

Individual evidence

1. ↑ Poincaré, H .: Sur les lignes géodésiques des surfaces convexes . Trans. Amer. Math. Soc. 6, 237-274 (1905)