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What is $\omega_j$ ? I remember the index of covariant tensor is in the upper right corner ,like this $\omega^j$. And $\omega$ always represent covariant tensor.So, I'm fuzzy with $\omega_j$.

In picture below ,is $\omega^i$ coefficient of $\omega$ ?

And what is$X^j$?

How to understand $g^{ij}\omega_j=\omega^i$?

enter image description here

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2 Answers 2

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Whenever you have a finite dimensional vector space $V$ and a metric $g$ on $V$, one can define an linear mapping $\flat: V\to V^*$, $X\mapsto X^\flat \in V^*$ given by $X^\flat (Y) = g(X, Y)$. Now let $\{e_1, \cdots, e_n\}$ be a basis of $V$, then every $X\in V$ is written as $X = X^i e_i$, or simply $X = X^i$.

Let $\{e^1, \cdots, e^n\}$ be its dual basis in $V^*$, that is $$\tag{1} e^i (e_j) = \delta^i_j\ \ \ \ \forall i, j = 1,\cdots, n,$$ Now to found out the representation of $X^\flat$ in terms of the basis $\{e^1, \cdots, e^n\}$, write $X^\flat = X_i e^i$, then by $(1)$, $$X_j = X^\flat (e_j) = g(X, e_j) = g(X^i e_i, e_j) = X^i g_{ij}.$$ Thus $X_j = g_{ij} X^i$.

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Usually lower indices represent the covariant components (of a covector) and upper indices the contravariant components ( of a vector) You can see here.

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  • $\begingroup$ How to understand $g^{ij}\omega_j=\omega^i$? $\endgroup$
    – Farmer
    Nov 14, 2015 at 1:37
  • $\begingroup$ It is the standard Einstein notation for raising and lovering indices ( here $g^{ij}$ is the metric tensor): en.wikipedia.org/wiki/Einstein_notation $\endgroup$ Nov 14, 2015 at 8:24

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