The metric tensor, using covariant components is $g_{ik}$ = $e^{(j)} \cdot e^{(k)}$ and using contravariant components it's $g^{ik}$ = $e_{(j)} \cdot e_{(k)}$. This seems counterintuitive to me. Why are the covariant components defined by the inner product of contravariant representations and vice versa? The text I'm reading doesn't go into a derivation and only says this occurs due to "[...] the transformation properties of the basis vectors when viewed from the standpoint of differential geometry." Can someone explain why the notation is like this?
2026-02-22 21:21:49.1771795309
Why are the covariant and contravariant components of the metric tensor defined this way?
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