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So, I've been thinking about this for a while...and I can't seem to resolve it in my head. In this thread I will use a tilde when referring to one forms and a vector sign when referring to vectors and boldface for tensors. It seems to me that if we require the basis vectors and one forms to obey the property that:

[tex]\tilde{\omega}^j(\vec{e}_i)=\delta_i^j[/tex]

Then, we cannot require the basis vectors and one forms to be "dual" to each other in the sense that we raise and lower their indices with the metric tensor. I.e.:

[tex]\tilde{\omega}^i=\bf{g}(\vec{e}_i,\quad)[/tex]

Since, unless my metric is the Cartesian metric, I cannot get the first property from the second quantity.

Is this true, or have I messed up somewhere?

[tex]\tilde{\omega}^j(\vec{e}_i)=\delta_i^j[/tex]

Then, we cannot require the basis vectors and one forms to be "dual" to each other in the sense that we raise and lower their indices with the metric tensor. I.e.:

[tex]\tilde{\omega}^i=\bf{g}(\vec{e}_i,\quad)[/tex]

Since, unless my metric is the Cartesian metric, I cannot get the first property from the second quantity.

Is this true, or have I messed up somewhere?

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