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A metric tensor is a nondegenerate symmetric covariant tensor that defines squared line elements by $d{s}^2=g_{\mu \nu }d{x}^{\mu }d{x}^{\nu }$.

A type $(0,s)$ covariant tensor is a multilinear form on $s$ vectors. For arbitrary $V_1,\dots ,V_s\in T_p\mathcal{M}$, the pullback of a covariant tensor isProjection of all covariant slots means $(P\omega )(X_1,\dots ,X_s)=\omega (P{X}_1,\dots ,P{X}_s)$. Since every pushed-forward vector is already tangential,Equality on all arguments proves ${\varphi }^{\ast }\omega ={\varphi }^{\ast }(P\omega )$. No antisymmetry is needed: this holds for every covariant tensor, not only for differential forms.