Geodesic

A geodesic (Pl. geodesics ), Geodetic also called geodesic or geodesic path is locally shortest curve between two points. Geodesics are solutions of an ordinary differential equation of second order, the geodesic equation.

Local and global definition

In Euclidean space geodesics are always straight. Relevant is the term " geodesic " only in curved spaces ( manifolds ), such as on a sphere or other curved surfaces or even in the curved spacetime of general relativity. One finds the geodesic lines with the help of the calculus of variations.

The restriction locally in the definition means that a geodesic only to be the shortest distance between two points needs if these points are close enough to each other; but it does not represent the global shortest path. Beyond the cutting place several geodesics of different lengths may lead to the same point, which prevents the global minimization of the length. For example, the shortest distance between two points on a sphere is always part of a great circle, but the two parts into which a great circle is divided by two points, are both geodesics, although only one of the two which is "globally" shortest connection.

Examples of geodesics of different rooms

• In the straight sections are exactly the geodesics.
• A geodesic on the sphere is always part of a great circle; it be based transcontinental flights and shipping routes (see geodesic ). All geodesic lines (or great circles ) on a sphere are self-contained - that is if you follow them, you eventually reaches the starting point again. On the other hand, this ellipsoid surfaces (which on the ellipsoid simple special cases of the geodesic are ) applies only along the meridians and the equator.
• In the special case of developable surfaces (eg, cone or cylinder), the geodesics are those curves in the settlement in the plane to be straight lines.

Classical differential geometry

In classical differential geometry, a geodesic is a path to an area in which all the principal normal coincides with the surface normal. This condition is only met if the geodesic curvature at each point is equal to 0.

Riemannian geometry

In Riemannian geometry, a geodesic is characterized by an ordinary differential equation. Let be a Riemannian manifold. A curve is called geodesic if the geodesic differential equation ( geodesic )

Met. It denotes the Levi- Civita connection. This equation means that the velocity vector field of the curve along the curve is constant. This definition is based on the consideration that the geodesics of just the straight lines and whose second derivative is constant zero.

Is a map of the manifold, we obtain with the help of the Christoffel symbols, the local representation

The geodesic equation. Here the Einstein summation convention is used. Those are the coordinate functions of the curve: The curve point has the coordinates.

From the theory of ordinary differential equations can be demonstrated that there is a unique solution of the geodesic differential equation with initial conditions and. And with the help of the first variation of, it can be shown that with respect to the Riemannian distance shortest curves satisfy the geodesic equation. Conversely, it can be shown that each geodesic at least locally, a shortest distance. That is on a geodesic there is a point at which the Geodetic no longer is the shortest connection. Is not compact the underlying manifold as the point can be infinite. One fixes a point and considered all at unit speed geodesic emanating from this point, it means the union of all intersections of the slice location. A geodesic with unit speed is a geodesic, applies.

In general, a geodesic must be defined on a time interval for a fitting. A Riemannian manifold is called geodesically complete if for each point and each tangent vector the geodesic is completely defined and on. The set of Hopf Rinow are several equivalent characterizations geodetically complete Riemannian manifolds.

Metric spaces

Be a metric space. For a curve, that is a continuous map, we define its length by

From the triangle inequality, the inequality follows.

As a minimizing geodesic in is denoted by a curve, ie a curve whose length realizes the distance between their endpoints. ( Geodesics in the sense of Riemannian geometry need not always be minimizing geodesics, but they are "local". )

A metric space is called geodesic metric space or length space if any two points can be connected by a minimizing geodesic. Complete Riemannian manifolds are length spaces. The Euclidean metric is with an example of a metric space, which is not a space length.