  A chiral alternative to the vierbein field in general relativity was
considered by 't Hooft in an attempt to facilitate the construction of a
quantum theory of gravity. These objects $ f^a{}_{\mu\nu}$ behave like the
"cube root" of the metric tensor. We try to construct specific representations
of these tensors in terms of Dirac $\gamma $ matrices in Euclidean and
Minkowski space, and promote these to curved space through Newman-Penrose
formalism. We conjecture that these are new objects with physical significance
and are the analog of Killing-Yano tensors. As an application, we try to
construct non-Abelian instanton like solutions in curved space from the flat
space 't Hooft tensors. The space part of the tensors are decomposed to a
product of Levi-Civita tensor and flat space Dirac gamma matrices. The gamma
matrices are promoted to the curved space with the help of vierbeins, as in
separability of Dirac equation by Chandrasekhar, using Newman-Penrose
formalism. The curved space generalisation of the instanton solution $A_{\mu}^a
$ is now constructed by substituting all objects to their general covariant
form.
