Why isn't Levi Civita defined for dimension unequal to rank? I can only find Levi-Civita defined for when rank (number of indices) is the same as the dimension (range of each index).
For example:

*

*$\epsilon_{ijk}$ is defined for $i \in \{1,2,3\}$, and

*$\epsilon_{ijkl}$ is defined for $i \in \{1,2,3,4\}$
Why can't I define some sort of $\epsilon_{ijk}$  for $i \in \{1,2,3,4\}$?
Is there some reason it would be ill defined, or is it just not useful for some reason?
 A: I would argue that the most useful general definition of the permutations symbols (sometimes referred to as the Levi-Civita symbols or alternating symbols) is through the  generalized Kronecker delta
$$\varepsilon^{h_1\dots h_n}=\delta^{j_1 \dots j_n}_{12\dots n}$$
$$\varepsilon_{j_1\dots j_n}=\delta^{12\ \dots n}_{j_1 \dots j_n}
$$
Where we draw attention to the fact that each of these quantities has exactly $n$ indices, where $n$ is the dimension of the underlying space (often a manifold).
Using this definition it is rather easy to show that $\varepsilon^{j_1\dots j_n}$ constitute the components of a type $(n,0)$ relative tensor of weight $+1$.
Similarly, it may be shown that $\varepsilon_{h_1\dots h_n}$ constitute the components of a type (0,n) relative tensor of weight $-1$.
Among all the beautiful and useful things this definition leads to there is a relation we will be using shortly
$$\tag{1}\label{usefulrelation}\varepsilon_{j_1\cdots j_n}\varepsilon^{h_1\cdots h_n}=\delta^{h_1\cdots h_n}_{j_1\dots j_n}
$$
Now back to your question. Should you try to extend this definition to a number of indices $<n$ you will probably end up with something that is not a) very useful and b) not tensorial.
For instance, at first glance it would only seem natural to consider the extension (for $n=3$)
$$
e^{ij}=\delta^{ij}_{12},\quad e_{hk}=\delta_{hk}^{12}
$$
But is this new entity tensorial? To study this we begin by stating the usual transformation law
$$\overline{\delta}^{ab}_{cd}=\frac{\partial \bar{x}^a}{\partial x^r}\frac{\partial \bar{x}^b}{\partial x^s}\frac{\partial x^u}{\partial \bar{x}^c}\frac{\partial x^w}{\partial \bar{x}^d}\delta^{rs}_{uw}
$$
with $c=1$ and $d=2$ using $(1)$
$$\overline{e}^{ab}=\frac{\partial \bar{x}^a}{\partial x^r}\frac{\partial \bar{x}^b}{\partial x^s}\frac{\partial x^u}{\partial \bar{x}^1}\frac{\partial x^w}{\partial \bar{x}^2}e^{rs}e_{uw}
$$
At this stage, under normal circumstances ($n=2$) we would use the relation $\frac{\partial x^u}{\partial \bar{x}^1}\frac{\partial x^w}{\partial \bar{x}^2}e_{uw}=J
$ where $J$ is the jacobian of the transformation to conclude that
$$\bar{e}^{ab}=J\frac{\partial \bar{x}^a}{\partial x^r}\frac{\partial \bar{x}^b}{\partial x^s}e^{rs}$$
Which is the transformation law for a type $(2,0)$ relative tensor of weight $+1$. However, since $n=3$ this identity no longer holds and we are stuck with
$$\bar{e}^{ab}=\left(\frac{\partial x^1}{\partial \bar{x}^1}\frac{\partial x^2}{\partial \bar{x}^2}-\frac{\partial x^2}{\partial \bar{x}^1}\frac{\partial x^1}{\partial \bar{x}^2}\right)\frac{\partial \bar{x}^a}{\partial x^r}\frac{\partial \bar{x}^b}{\partial x^s}e^{rs}=J ^{33}\frac{\partial \bar{x}^a}{\partial x^r}\frac{\partial \bar{x}^b}{\partial x^s}e^{rs}$$
Where $J^{33}$ is the cofactor of the element $\frac{\partial x^3}{\partial \bar{x}^3}$ in the Jacobian. Of course we could invent a new breed of tensors called the cofactorian tensors, but this would be a highly inconvenient transformation law with no obvious generalizations.
