# Geodesic

straight path on a curved surface or a Riemannian manifold

In geometry and physics, a geodesic is the shortest curve, confined to a surface (or higher-dimensional manifold), joining two points on the surface (or higher-dimensional manifold). On a two-dimensional sphere, the geodesics are called great circle arcs.

## Quotes

• If our geometry is to resemble differential geometry we must adjoin some uniqueness properties. Now in those geometries the geodesics, and more generally the externals in the calculus of variations, are given by differential equations of the second order, and under the hypotheses usually made in those fields, there is just one solution through a given line element. Thus a geodesics has a unique prolongation, though the shortest geodesic are joining two points even on simple surfaces such as the sphere, need not be unique.
• A geodesic that is not a null geodesic has the property that  ds, taken along a section of the track with the end points P and Q, is stationary if one makes a small variation of the track keeping the end points fixed.
• We begin by recalling that geodesics can be obtained as solutions of the Euler-Lagrange equation of a Lagrangian given by the kinetic energy. We define symplectic and contact manifolds and we set up the basic geometry of the tangent bundle; we introduce the connection map, horizontal and vertical subbundles, the Sasaki metric, the symplectic form and the contact form. We describe the main properties of these objects and we show that the geodesic flow is a Hamiltonian flow. Also, when we restrict the geodesic flow to the unit sphere bundle of the manifold, we obtain a contact flow. The contact form naturally induces a probability measure that is invariant under the geodesic flow and is called the Liouville measure.
• The idea of a parallel displacement along some given curve in a two-dimensional surface can be given an intuitive interpretation. Suppose the surface is developable. Then we can unroll it on to a plane and parallel-displace vectors in the plane. The surface is then rolled back and we have the required parallel-transported vector. If a given surface is not developable, we must first select a path for parallel transport, then erect a tangent plane at each point of the path. These tangent planes will envelope a developable surface. This new developable surface can then be unrolled and the operations of parallel transport and rerolling carried out. If the curve along which the parallel displacement is to be carried out happens to be a geodesic, it becomes a straight line when unrolled on to a plane. It is then clear that the angle between a geodesic and a vector remains unchanged in a parallel displacement.