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. Definition 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. Examples Killing field on the circle 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. Killing fields in flat space Here we derive the Killing fields for general flat space. From Killing's equation and the Ricci identity for a covector , (using abstract index notation) where is the Riemann curvature tensor, the following identity may be proven for a Killing field : When the base manifold is flat space, that is, Euclidean space or possibly Pseudo-Euclidean space (as for Minkowski space), we can choose global flat coordinates such that in these coordinates, the Levi-Civita connection and hence Riemann curvature vanishes everywhere, giving Integrating and imposing the Killing equation allows us to write the general solution to as where is antisymmetric. By taking appropriate values of and , we get a basis for the generalised Poincaré algebra of isometries of flat space: These generate pseudo-rotations (rotations and boosts) and translations respectively. Intuitively these preserve the (pseudo)-metric at each point. For (Pseudo-)Euclidean space of total dimension, in total there are generators, making flat space maximally symmetric. This number is generic for maximally symmetric spaces. Maximally symmetric spaces can be considered as sub-manifolds of flat space, arising as surfaces of constant proper distance which have O(p,q) symmetry. If the submanifold has dimension , this group