Projective vector field


A projective vector field is a smooth vector field on a semi Riemannian manifold whose flow preserves the geodesic structure of without necessarily preserving the affine parameter of any geodesic. More intuitively, the flow of the projective maps geodesics smoothly into geodesics without preserving the affine parameter.

Decomposition

In dealing with a vector field on a semi Riemannian manifold, it is often useful to decompose the covariant derivative into its symmetric and skew-symmetric parts:
where
and
Note that are the covariant components of.

Equivalent conditions

Mathematically, the condition for a vector field to be projective is equivalent to the existence of a one-form satisfying
which is equivalent to
The set of all global projective vector fields over a connected or compact manifold forms a finite-dimensional Lie algebra denoted by and satisfies for connected manifolds the condition:. Here a projective vector field is uniquely determined by specifying the values of, and at any point of. Projectives also satisfy the properties:

Subalgebras

Several important special cases of projective vector fields can occur and they form Lie subalgebras of. These subalgebras are useful, for example, in classifying spacetimes in general relativity.

Affine algebra

s satisfy and hence every affine is a projective. Affines preserve the geodesic structure of the semi Riem. manifold whilst also preserving the affine parameter. The set of all affines on forms a Lie subalgebra of denoted by and satisfies for connected M,. An affine vector is uniquely determined by specifying the values of the vector field and its first covariant derivative at any point of. Affines also preserve the Riemann, Ricci and Weyl tensors, i.e.

Homothetic algebra

s preserve the metric up to a constant factor, i.e.. As, every homothety is an affine and the set of all homotheties on forms a Lie subalgebra of denoted by and satisfies for connected M
A homothetic vector field is uniquely determined by specifying the values of the vector field and its first covariant derivative at any point of the manifold.

Killing algebra

preserve the metric, i.e.. Taking in the defining property of a homothety, it is seen that every Killing is a homothety and the set of all Killing vector fields on forms a Lie subalgebra of denoted by and satisfies for connected M
A Killing vector field is uniquely determined by specifying the values of the vector field and its first covariant derivative at any point of.

Applications

In general relativity, many spacetimes possess certain symmetries that can be characterised by vector fields on the spacetime. For example, Minkowski space admits the maximal projective algebra, i.e..
Many other applications of symmetry vector fields in general relativity may be found in Hall which also contains an extensive bibliography including many research papers in the field of symmetries in general relativity.