At the end of page 17 in the book General Relativity for Mathematicians of Sachs, Wu, the author writes
For two vector fields $X$ and $Y$, the tensor fields physically equivalent to $X\otimes Y$ are precisely: $g(X,\cdot)\otimes Y,\ X\otimes g(Y,\cdot)$ and $g(X,\cdot)\otimes g(Y,\cdot)$.
Is I understand, the notation $X\otimes Y$ refers to the tensor product of $X$ and $Y$, but this means $X,\, Y$ must be the tensors over $T_xM$. But in this context, the author presumed that $X$ and $Y$ are vector fields on $M$, which mean output of $X$ and $Y$ is not a scalar, but a vector. So what is the product $X\otimes Y$ considered here ? I'm confused here, hope someone will help me clarify this. Thanks