I have a confusion lowering and raising in Levi-Civita indices. I don't understand which of the following relation is correct. $$\epsilon^{ijk} = \epsilon_{ijk}$$ or $$\epsilon^{ijk} = - \epsilon_{ijk}$$
Where I am using metric $ diag(1.-1) $. Also tell me if there is some link between metric tensor and Levi-Civita tensor. I am putting a snap from some book on covariant formalism of electromagnetism.
On
It looks like you're using the metric $dt^2 - dx^2 - dy^2 - dz^2$? If so, you change sign every time you raise/lower a spacelike index in Minkowski coordinates - for example, your screenshot includes the equation $B_k = -B^k$. Since $\epsilon$ has three indices, raising/lowering all three changes the sign by $(-1)^3 = -1$.
From wikipedia: In a context where tensor index notation is used to manipulate tensor components, the Levi-Civita symbol may be written with its indices as either subscripts or superscripts with no change in meaning, as might be convenient. Thus, one could write
$$\varepsilon^{ij\dots k}=\varepsilon_{ij\dots k} $$
You can see that it is not called "tensor", in which case it would need the metric tensor to raise/lower its indices, but "symbol". It is just a synthetic way to express a collection of $0, 1, -1$ depending of the value and order of the indeces.