We consider classical gauge theory with spontaneous symmetry breaking on a principal bundle $Pto X$ whose structure group $G$ is reducible to a closed subgroup $H$, and sections of the quotient bundle $P/Hto X$ are treated as classical Higgs fields. Its most comprehensive example is metric-affine gauge theory on the category of natural bundles where gauge fields are general linear connections on a manifold $X$, classical Higgs fields are arbitrary pseudo-Riemannian metrics on $X$, and matter fields are spinor fields. In particular, this is the case of gauge gravitation theory.
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