Jacobi field

A Jacobi field or, more precisely Jacobi vector field is a vector field along a geodesic, which is solution of the Jacobi equation. Considered Clearly it represents the displacement vector field between infinitesimally close geodesics on a Riemannian or pseudo - Riemannian manifold dar. use is this concept in differential geometry and general relativity.

Jacobi equation

Jacobi equation is an equation for a vector field along a geodetic line and sets the curvature of the manifold in relation to the second derivative of the Jacobi this field.

A geodesic with Tangentialvektorfeld c 'and is a smooth vector field along c, then J is a Jacobian field if the Jacobi equation

Met. It denotes the curvature tensor and the induced by the Levi- Civita connection covariant derivative.

Generally, the Jacobi field components can have tangential and orthogonal to TM for Tangentialvektorfeld the geodesic. It is along the geodesic c (t ) for a Jacobi field J: the vector field Y = Y ac ' btc ' is for real parameters a, b ​​also a Jacobi field along c. For the investigation of conjugate points and the Morse index theory is therefore usually limited to the consideration of the orthogonal component by considering equivalence classes modulo tangential components.

Example: Jacobi field on the 2-sphere

As an illustrative example, the spherical surface can be used. In the canonical metric on the sphere which one wins in the embedding, the geodesics are the great circles. All great circles which pass through a point intersect again at the antipodal point of this point. These great circles together with the two points thus describe the longitudes and the poles of a spherical coordinate system. The Jacobi field along these meridians at any point tangent to the sphere and perpendicular to the meridian. The integral curves of the Jacobi field are therefore the parallels. The lines of latitude lines in the picture can be seen as vectors of the Jacobi field understand at this point. Here it is clear that the Jacobi field describes the distance between neighboring geodesics and disappears into the poles. The two poles are thus mutually conjugate points.

Lorentz index form and conjugate points

The Lorentz index theory regarded as the Riemann index theory geodesics on a special case of pseudo- Riemannian manifolds and studied these geodesics in the presence of conjugate points. Two points along a geodesic is called conjugate to each other, if a non-trivial smooth Jacobi field exists along which disappears in and.

Be the space of smooth sections orthogonal vector fields along a geodesic, that is, for his and for all. The symmetric bilinear form of index

Is defined by

In the Riemann index theory the sign of the index form is chosen positive. If smooth is, a partial integration be carried out and we have:

For with, that is, to this further simplifies to:

The index form is closely related to conjugate points: For equivalent if a Jacobi field is whether or applies to everyone. So the endpoints along the geodesics are then conjugated exactly when the bilinear form is degenerate.

Variation of geodesics

A variation of a geodesic is a smooth map

For a with. Usually, calls still fixed endpoints: and for all. The canonical variation with fixed endpoints is now just the exponential of scaled with s vector fields

The variation vector field V of variation is the vector field V ( t) along c (t). For the canonical variation, the variation vector field is therefore Y ( t).

The second variation of the Lorentz length between

The geodesic variation is now given by the index form described above: . As a result, the variation in I (Y, Y) results in > 0 adjacent time-like curves, which also connect C (a) with c ( B ), but a larger length

Have. Thus, the time-like geodesic is maximal, ie their length corresponds to the distance between Lorentz their endpoints, must I be on negative semidefinite c.