Affine vector field
Encyclopedia
An affine vector field is a projective vector field
preserving geodesic
s and preserving the affine parameter. Mathematically, this is expressed by the following condition:
Projective vector field
A projective vector field is a smooth vector field on a semi Riemannian manifold M whose flow preserves the geodesic structure of M without necessarily preserving the affine parameter of any geodesic...
preserving geodesic
Geodesic
In mathematics, a geodesic is a generalization of the notion of a "straight line" to "curved spaces". In the presence of a Riemannian metric, geodesics are defined to be the shortest path between points in the space...
s and preserving the affine parameter. Mathematically, this is expressed by the following condition:
See also
- Conformal vector field
- Curvature collineation
- Homothetic vector field
- Killing vector fieldKilling vector fieldIn mathematics, a Killing vector field , named after Wilhelm Killing, is a vector field on a Riemannian manifold that preserves the metric. Killing fields are the infinitesimal generators of isometries; that is, flows generated by Killing fields are continuous isometries of the manifold...
- Matter collineation
- Spacetime symmetriesSpacetime symmetriesSpacetime symmetries are features of spacetime that can be described as exhibiting some form of symmetry. The role of symmetry in physics is important in simplifying solutions to many problems, spacetime symmetries finding ample application in the study of exact solutions of Einstein's field...