how we can add a 2-tensor to a 1-form?

Question

In the definition of a Randere norm, we add a Riemannian metric $\alpha$ to a 1-from $\beta$. Indeed, $F(y)=\sqrt{a_{ij}(x)y^iy^j}+b_i(x)y^i$ in ehich $\alpha(y)=\sqrt{a_{ij}(x)y^iy^j}$ is a Riemannian norm and $\beta(y)=b_i(x)y^i$ a 1-form. Here $x\in M$ and $y\in T_xM$.

Now I am wondering how we are allowed to add a 2-tensor (Riemannian metric) to a 1-form.

Answer 1

The term $\sqrt{a_{ij}(x) y^i y^j}$ is not a 2-tensor since it is not linear. It's just a number.

The other term, $b_i(x) y^i,$ is also just a number. It's $b = b_i(x) \, \mathrm{d}x^i$ that is a 1-form.

I think that the construction is quite clear in this paper.