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Here ''R''<sub>''g''</sub> denotes the right action of ''G'' on ''P'' for some ''g'' ∈ ''G''. Note that for 0-forms the second condition is [[vacuously true]].
Example: If ρ is the [[adjoint representation]] of ''G'' on the Lie algebra, then the connection form ω satisfies the first condition (but not the second). The associated [[curvature form]] Ω satisfies both; hence Ω is a tensorial form of adjoint type. The "difference" of two connection forms is a tensorial form.
Given ''P'' and ''ρ'' as above one can construct the [[associated vector bundle]] ''E'' = ''P'' ×<sub>''ρ''</sub> ''V''. Tensorial ''q''-forms on ''P'' are in a natural one-to-one correspondence with ''E''-valued ''q''-forms on ''M''. As in the case of the principal bundle F(''E'') above, given a ''q''-form <math>\overline{\phi}</math> on ''M'', define φ on ''P'' fiberwise by, say at ''u'',
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