How to change the indice order of a rank 3 tensor?

Question

I am new to tensor analysis,

Suppose I have a contravariant Tensor say $X^{ijk}$, in order to do the indices contraction, I may have to calculate $X^{jik}g_{j} = X^{ik}$ (Einstein summation omitted here), then how should I generate the $X^{jik}$ from $X^{ijk}$? Is there any rule for that? is that symmetrical principle?

Thanks very much.

Answer 1

There is no relation whatsoever between $X^{ijk}$ and $X^{jik}$ as stated. A given tensor need not be symmetric or skew-symmetric for any choices of indices.

Also, your $X^{jik}g_j = X^{ik}$ is not correct, as the metric tensor is a tensor of order $2$. What would make sense is to write $X^{jik}g_{k\ell} = X^{ji}_{\phantom{ji}\ell}$ instead.

For better or worse, I have some notes about tensors on vector spaces, you might find that useful.