Q: Metric connection as map from $\Omega^1 \, \to \, \Omega^1 \otimes \, \Omega^1$ I was reading Shahn Majid's and Edwin Begg's book on Quantum Reimannian Geometry. In this they define the metric to be a map $$g:\Omega^1 \otimes \Omega^1 \to A.$$
I do not understand this definition. In GR, the metric was a $(0,\,2)$ tensor that took two vectors and gave a scalar output. What merits this redefinition? Or are these equivalent.
Furthermore, this brings us to define the connection as a map $$ \nabla:\,\Omega^1 \, \to\,  \Omega^1\otimes \Omega^1.$$ How does this work? Should it not be a map from $(0, n)$ tensors to $(0,n+1)$ tensors?
Sorry for not clarifying this before: $\Omega^1$ is the space of 1-forms over $A=\mathbb{C}(X)$ (which is the algebra of complex valued, finitely supported (admits a basis of Kronecker deltas) functions over $X$, which is a finite set. There exists the exterior derivative map $d:A\to \Omega^1$.
