Coordinate independent expression for transverse trace-free tensors

The transverse and trace-free (TT) part of the extrinsic curvature represents half of the dynamical degrees of freedom of the gravitational field in the $3 + 1$ formalism. As such, it is part of the freely specifiable initial data for numerical relativity. Though TT tensors in 3-space possess only two component degrees of freedom, they cannot ordinarily be given solely by two scalar potentials. Such expressions have been derived however, in coordinate form, for all TT tensors in flat space which are also translationally or axially symmetric (Conboye and \'O Murchadha 2014 \emph{Class. Quantum Grav.} {\bf 31} 085019). Since TT tensors are conformally covariant, these also give TT tensors in conformally-flat space. In this article, the work above has been extended by giving a coordinate-\emph{independent} expression for these TT tensors. The translational and axial symmetry conditions have also been generalized to invariance along any hypersurface orthogonal Killing vector.