General covariant transformations

In physics, general covariant transformations are symmetries of gravitation theory on a world manifold formula_1. They are gauge transformations whose parameter functions are vector fields on formula_1. From the physical viewpoint, general covariant transformations are treated as particular (holonomic) reference frame transformations in general relativity. In mathematics, general covariant transformations are defined as particular automorphisms of so-called natural fiber bundles. 
Mathematical definition.
Let formula_3 be a fibered manifold with local fibered coordinates formula_4. Every automorphism of formula_5 is projected onto a diffeomorphism of its base formula_1. However, the converse is not true. A diffeomorphism of formula_1 need not give rise to an automorphism of formula_5.
In particular, an infinitesimal generator of a one-parameter Lie group of automorphisms of formula_5 is a projectable vector field
on formula_5. This vector field is projected onto a vector field formula_12 on formula_1, whose flow is a one-parameter group of diffeomorphisms of formula_1. Conversely, let formula_15 be a vector field on formula_1. There is a problem of constructing its lift to a projectable vector field on formula_5 projected onto formula_18. Such a lift always exists, but it need not be canonical. Given a connection formula_19 on formula_5, every vector field formula_18 on formula_1 gives rise to the horizontal vector field
on formula_5. This horizontal lift formula_25 yields a monomorphism of the formula_26-module of vector fields on formula_1 to the formula_28-module of vector fields on formula_5, but this monomorphisms is not a Lie algebra morphism, unless formula_19 is flat.
However, there is a category of above mentioned natural bundles formula_31 which admit the functorial lift formula_32 onto formula_33 of any vector field formula_18 on formula_1 such that formula_36 is a Lie algebra monomorphism
This functorial lift formula_32 is an infinitesimal general covariant transformation of formula_33.
In a general setting, one considers a monomorphism formula_40 of a group of diffeomorphisms of formula_1 to a group of bundle automorphisms of a natural bundle formula_31. Automorphisms formula_43 are called the general covariant transformations of formula_33. For instance, no vertical automorphism of formula_33 is a general covariant transformation.
Natural bundles are exemplified by tensor bundles. For instance, the tangent bundle formula_46 of formula_1 is a natural bundle. Every diffeomorphism formula_48 of formula_1 gives rise to the tangent automorphism formula_50 of formula_46 which is a general covariant transformation of formula_46. With respect to the holonomic coordinates formula_53 on formula_46, this transformation reads
A frame bundle formula_56 of linear tangent frames in formula_46 also is a natural bundle. General covariant transformations constitute a subgroup of holonomic automorphisms of formula_56. All bundles associated with a frame bundle are natural. However, there are natural bundles which are not associated with formula_56.