Considering a class of diffeomorphisms which are not isometries but change metrics a tiny bit, with some asymptotic boundary conditions.
This is useful to obtain conformal symmetry algebras in some physical theories of gravity, esp. in 3D.
J. Brown, M. Henneaux, Central charges in the canonical realization of asymptotic symmetries: an example from three-dimensional gravity, Comm. Math. Phys. 104 (1986) 207 (key paper!!) euclid
Coussaert, M. Henneaux, P. van Driel, Class. Quant. Grav. 12 (1995) 2961
…Brown-Henneaux boundary condition