of symmetries has the expected dimension (as a Lie group). Heuristically, we can derive the dimension of the Killing field algebra. Treating Killing's equation together with the identity as a system of second order differential equations for , we can determine the value of at any point given initial data at a point . The initial data specifies and , but Killing's equation imposes that the covariant derivative is antisymmetric. In total this is independent values of initial data. For concrete examples, see below for examples of flat space (Minkowski space) and maximally symmetric spaces (sphere, hyperbolic space). Killing fields on the hyperbolic plane 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 . Killing fields on a 2-sphere The Killing fields of the two-sphere , or more generally the -sphere should be "obvious" from ordinary intuition: spheres, having rotational symmetry, should possess Killing fields which generate rotations about any axis. That is, we expect to have symmetry under the action of the 3D rotation group SO(3). When explicitly expressed in terms of the standard coordinate chart for , the Killing fields have a non-obvious structure that obscures their nature. This is articulated below. This "non-obvious" structure is generic to manifolds that are not spheres, and thus the 2-sphere provides a good toy model on which to explore the intuitive interpretation of Killing fields. The conventional chart for the 2-sphere embedded in in Cartesian coordinates is given by so that parametrises the height, and parametrises rotation about the -axis. Pulling back the standard Cartesian metric to these coordinates gives the standard metric on the sphere,