It's been a while since I studied differential geometry, so I forgot a lot of basic things. I got therefore stuck at a specific sentence in Milnor's book on complex dynamics.
It says the following: A Riemannian metric on an open subset of $\mathbb{C}$ can be described as an expression of the form $$ ds^2 = g_{11} dx^2 + 2g_{12}dxdy + g_{22}dy^2$$, where $[g_{jk}]$ is a positive definite matrix, which depends smoothly on the point $z=x +iy$.
I'm confused about the $2g_{12}dxdy$ part. Because, I thought that the Riemannian metric should be symmetric on the two tangent vectors it gets as input. And if I'm not mistaken, $dxdy$ basically means $dx \otimes dy$ and gets two vectors as input. But $dx \otimes dy (v,w) \neq dx \otimes dy(w,v)$.
I doubt that it's a mistake from the book, so I'm wondering what I misunderstood. Thank you. PS I hope it's not a duplicate (at least it doesn't seem so for me)
In this context, $dxdy$ has to be understood as the symmetric product of $dx$ and $dy$, sometimes denoted by $dx\cdot dy$ or $dx \odot dy$, and which is equal to $\frac{1}{2}(dx\otimes dy + dy \otimes dx)$.