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## Summary:

- For a non-coordinate basis, because there is no corresponding coordinate system, then the definition of partial derivative and a closed path is in question.

## Main Question or Discussion Point

We always can define a metric with a basis field ##g_{\mu\nu}=e_\mu \cdot e_\nu##, For a basis field ##e_\mu##, it can belong to a coordinate basis, then there is a corresponding coordinate system##\{x^\mu\}##,then we can have ##e_\mu=\frac{\partial}{\partial x^\mu}##, and ##[e_\mu , e_\nu]=0## ,but for a non-coordinate system, there is no corresponding coordinate system. then we will find that:

(1) If the metric is defined with a non-coordinate system, then there is no cooresponding coordinate system, then it seems that we will have trouble defining the partial derivative ##\partial_\mu##:

(2) To define curvature we should parallel a vector along a closed path in the space. In a coordinate system, it is very easy to describe a closed path. For example we can define a parallelogram which contains four infinitesimal segments: ##\epsilon e_a , \epsilon e_b , -\epsilon e_a , -\epsilon e_b ##. But in a space only equipped with a non-coordinate basis, because ##[ e_\mu ,e_\nu]\neq 0##, so the path ##\epsilon e_\mu , \epsilon e_\nu , -\epsilon e_\mu , -\epsilon e_\nu ## is not a closed path, so how to define a closed path?

(1) If the metric is defined with a non-coordinate system, then there is no cooresponding coordinate system, then it seems that we will have trouble defining the partial derivative ##\partial_\mu##:

(2) To define curvature we should parallel a vector along a closed path in the space. In a coordinate system, it is very easy to describe a closed path. For example we can define a parallelogram which contains four infinitesimal segments: ##\epsilon e_a , \epsilon e_b , -\epsilon e_a , -\epsilon e_b ##. But in a space only equipped with a non-coordinate basis, because ##[ e_\mu ,e_\nu]\neq 0##, so the path ##\epsilon e_\mu , \epsilon e_\nu , -\epsilon e_\mu , -\epsilon e_\nu ## is not a closed path, so how to define a closed path?