Special invariant tensors

There are two special tensors, which components are invariant under arbitrary coordinate transformations.

The first one is rank-2 unit tensor that is represented by the unit matrix

$${\delta^{alpha}}_\beta \equiv 1 if \alpha=\beta, \quad 0, if \alpha \ne \beta$$ (11)

which is often called Kronecker symbol. Note that it has mixed components. One may encounter $\delta_{\alpha \beta}$ which in some coordinate system is represented by unit matrix, but such tensor will have its components changed in another frame (check !). The effect of contraction of the Kronecker symbol with another vector is the replacement of the component index

$${\delta^\alpha}_\beta A^\beta = A^\alpha, \quad\quad {\delta^\alpha}_\beta A^\alpha = A^\beta$$ (12)

The second special tensor has rank equal to dimensionality $N$ of the space, and is defined as

$$\varepsilon^{\alpha\beta\gamma \ldots} = \pm 1 \quad if and only if \quad \alpha \ne \beta \ne \gamma \ldots$$ (13)

with $e^{012\ldots}=1$ and changing sign with each permutation of a pair of indexes. Thus, $e^{102\ldots}=-1$, etc. This tensor is known as fully antisymmetric tensor of rank $N$ or Levi-Civita symbol. With its help one can define a dual tensor to any fully antisymmetric tensor of rank $r$ less than $N$. Such dual tensor will have rank $N-r$. For example in 4D if $F_{\gamma\delta}$ is antisymmetric, its dual is

$$(F^*)^{\alpha\beta} = \varepsilon^{\alpha\beta\gamma\delta} F_{\gamma\delta}$$ (14)

while for a tensor of rank 3, the dual will be a vector

$$(A^*)^{\alpha} = \varepsilon^{\alpha\beta\gamma\delta} A_{\beta\gamma\delta}$$ (15)