Geodesics

Shortest connection (geodesic) on the globe

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

Local and global definition

In Euclidean space , geodesics are always straight lines . The term “geodesic” is only relevant in curved spaces ( manifolds ), such as on a spherical surface or other curved surfaces or in the curved spacetime of general relativity . The geodetic lines are found with the help of the calculus of variations .

The local restriction in the definition means that a geodesic only needs to be the shortest connection between two points if these points are close enough to one another; however, it does not have to represent the shortest path globally . Beyond the intersection , several geodesics of different lengths can lead to the same point, which prevents the global minimization of the length. For example, the shortest connection between two non- antipodal points on a sphere is always part of a unique great circle , but the two parts into which this great circle is divided by these two points are both geodesics, although only one of the two represents the globally shortest connection .

Examples of geodesics of different spaces

• In with Euclidean metric are exactly straight sections Geodetic. ${\displaystyle \mathbb {R} ^{n}}$
• A geodesic on the sphere is always part of a great circle ; transcontinental flight and shipping routes are based on this (see Orthodrome ). All geodetic lines (or great circles) on a sphere are self-contained - that is, if you follow them, you will eventually reach the starting point again. On ellipsoid surfaces, however, this only applies along the meridians and the equator (which are simple special cases of the geodetic line on the ellipsoid).
• In the special case of developable surfaces (e.g. cones or cylinders ), the geodesics are those curves that become straight lines when developed into the plane.

Classic differential geometry

Geodetic (red) in a two-dimensional , curved space that is embedded in a three-dimensional space. (Modeling of gravity using the geodesics in relativity theory)

In classical differential geometry , a geodetic is a path on a surface in which the main normal coincides with the surface normal everywhere . This condition is fulfilled if and only if the geodetic curvature is equal to 0 at every point . ${\displaystyle \gamma \colon I\to S}$ ${\displaystyle S\subset \mathbb {R} ^{3}}$

Riemannian geometry

In Riemannian geometry , a geodetic is characterized by an ordinary differential equation . Let be a Riemannian manifold . A curve is called geodesic if it contains the geodetic differential equation ( geodesic equation ) ${\displaystyle M}$ ${\displaystyle \gamma \colon I\to M}$

${\displaystyle \nabla _{\dot {\gamma }}{\dot {\gamma }}=0}$

Fulfills. It denotes the Levi-Civita connection . This equation means that the velocity vector field of the curve is constant along the curve. This definition is based on the consideration that the geodetic des are precisely the straight lines and their second derivative is constant zero. ${\displaystyle \nabla }$ ${\displaystyle \mathbb {R} ^{n}}$

If there is a map of the manifold, the local representation is obtained with the help of the Christoffel symbols ${\displaystyle (U,x)}$ ${\displaystyle \Gamma _{kl}^{m}}$

${\displaystyle {\ddot {x}}^{m}+\Gamma _{kl}^{m}{\dot {x}}^{k}{\dot {x}}^{l}=0}$

the geodetic differential equation. Here is the Einstein notation used. The the coordinates functions of the curve : the curve point has coordinates . ${\displaystyle x^{m}}$ ${\displaystyle \gamma }$ ${\displaystyle \gamma (t)}$ ${\displaystyle (x^{1}(t),\dots ,x^{n}(t))}$

From the theory of ordinary differential equations it can be shown that there is a unique solution of the geodetic differential equation with the initial conditions and . And with the help of the first variation of it can be shown that the curves that are shortest with regard to the Riemannian distance satisfy the geodetic differential equation. Conversely, one can show that every geodetic is at least locally a shortest connection. That means, on a geodetic there is a point from which the geodetic is no longer the shortest connection. If the underlying manifold is not compact , the point can also be infinite. If one fixes a point and looks at all geodesics with unit speed, which emanate from this point, then the union of all intersection points is called the intersection location . A geodesic with unit velocity is a geodesic for which applies. ${\displaystyle \gamma (t_{0})=p}$ ${\displaystyle {\dot {\gamma }}(t_{0})=V\in T_{p}M}$ ${\displaystyle \gamma }$ ${\displaystyle d(.,.)}$ ${\displaystyle \gamma }$ ${\displaystyle \|{\dot {\gamma }}\|=1}$

In general, a geodesic only needs to be defined on a time interval for a suitable one. A Riemannian manifold is called geodetically complete if for every point and every tangential vector the geodesic with and on is completely defined. The Hopf-Rinow theorem gives various equivalent characterizations of geodetically complete Riemannian manifolds. ${\displaystyle (-\epsilon ,\epsilon )}$ ${\displaystyle \epsilon >0}$ ${\displaystyle p\in M}$ ${\displaystyle v\in T_{p}M}$ ${\displaystyle \gamma }$ ${\displaystyle \gamma (0)=p}$ ${\displaystyle {\dot {\gamma }}(0)=v}$ ${\displaystyle \mathbb {R} }$

In general, a geodesic (in the sense of Riemannian geometry defined above) is only locally minimizing, but not globally. In other words, it does not necessarily have to be the shortest connection between and for everyone , but there is one , so that the shortest connection between and is for everyone . ${\displaystyle \gamma }$ ${\displaystyle \gamma (0)}$ ${\displaystyle \gamma (t)}$ ${\displaystyle t}$ ${\displaystyle \delta >0}$ ${\displaystyle \gamma }$ ${\displaystyle t\in \left[-\delta ,\delta \right]}$ ${\displaystyle \gamma (0)}$ ${\displaystyle \gamma (t)}$

A geodesic is called minimizing geodesic if the shortest connection between and is for all . ${\displaystyle \gamma }$ ${\displaystyle t}$ ${\displaystyle \gamma (0)}$ ${\displaystyle \gamma (t)}$

Metric spaces

→ Main article : Geodetic metric space

Be a metric space . For a curve, that is, a continuous mapping , its length is defined by ${\displaystyle (X,d)}$ ${\displaystyle \gamma \colon \left[a,b\right]\rightarrow X}$

${\displaystyle L(\gamma )=\sup _{a=t_{0}<t_{1}<\cdots <t_{n}=b}\sum _{i=0}^{n-1}d(\gamma (t_{i}),\gamma (t_{i+1}))}$.

The inequality follows from the triangle inequality . ${\displaystyle L(\gamma )\geq d(\gamma (a),\gamma (b))}$

As minimizing geodesic in refers to a curve with , that is a curve whose length realizes the distance between their endpoints. (Geodesics in the sense of Riemannian geometry do not always have to be minimizing geodesics, but they are "local".) ${\displaystyle (X,d)}$ ${\displaystyle \gamma \colon \left[a,b\right]\rightarrow X}$ ${\displaystyle L(\gamma )=d(\gamma (a),\gamma (b))}$

A metric space is called geodetic metric space or length space if two points can be connected by a minimizing geodesic. Complete Riemannian manifolds are length spaces. The one with the Euclidean metric is an example of a metric space that is not a length space. ${\displaystyle (X,d)}$ ${\displaystyle \mathbb {R} ^{2}\setminus \left\{(0,0)\right\}}$

literature

• Manfredo Perdigão do Carmo: Riemannian geometry. Birkhäuser, Boston et al. 1992, ISBN 0-8176-3490-8 .

Web links

• Geodesics on parameterized areas - sage interact - Interactive Sage -worksheet that calculates and visualizes geodesics on parameterized areas.