Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.
Concept
Killing vector field
Summary
In mathematics, a Killing vector field (often called a Killing field), named after Wilhelm Killing, is a vector field on a Riemannian manifold (or pseudo-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. More simply, the flow generates a symmetry, in the sense that moving each point of an object the same distance in the direction of the Killing vector will not distort distances on the object.
Specifically, a vector field X is a Killing field if the Lie derivative with respect to X of the metric g vanishes:
In terms of the Levi-Civita connection, this is
for all vectors Y and Z. In local coordinates, this amounts to the Killing equation
This condition is expressed in covariant form. Therefore, it is sufficient to establish it in a preferred coordinate system in order to have it hold in all coordinate systems.
The vector field on a circle that points clockwise and has the same length at each point is a Killing vector field, since moving each point on the circle along this vector field simply rotates the circle.
A toy example for a Killing vector field is on the upper half-plane equipped with the Poincaré metric . The pair is typically called the hyperbolic plane and has Killing vector field (using standard coordinates). This should be intuitively clear since the covariant derivative transports the metric along an integral curve generated by the vector field (whose image is parallel to the x-axis).
Furthermore, the metric is independent of from which we can immediately conclude that is a Killing field using one of the results below in this article.
The isometry group of the upper half-plane model (or rather, the component connected to the identity) is (see Poincaré half-plane model), and the other two Killing fields may be derived from considering the action of the generators of on the upper half-plane. The other two generating Killing fields are dilatation and the special conformal transformation .
Official source
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.