pseudovectors and covectors

Question

The cross product of two vectors, $\vec{c}=\vec{a}\times\vec{b}$, is sometimes characterized as a pseudovector or axial vector.

On the other hand, if we look at the vectors $\vec{a}$ and $\vec{b}$ as components of 1-forms, then the elemens of the cross product can be seen as components of a 2-form, and $\vec{c}$ should be seen as the Hodge dual of this 2-form. This, to me, is the real nature of pseudovectors.

My question is whether $\vec{c}$ can be seen as a covector, or a covariant 1-tensor. That is, if I write $a^i$ for the contravariat components of $\vec{a}$ and similar for $\vec{b}$, is there a tensor with components $T_{i,j,k}$ such that the covariant components of $\vec{c}$ are given by $c_k= T_{i,j,k}a^ib^j$?

Answer 1

This paper is relevant; it shows that the Levi-Civita symbol becomes a tensor when multiplied by the square root of the modulus of the determinant of the metric tensor.

Answer 2

In general, covectors and pseudo-vectors are completely different concepts. Both come from some notion of duality.

If $V$ is a real vector space, the the dual space is the space of real linear functionals on $V$. Such things are called covectors. This notion of duality does not depend on the dimension of $V$.

There is another kind of duality which is Hodge duality. In $\mathbb{R}^n$, this associates totally antisymmetric tensors of rank $k$ with totally antisymmetric tensors of rank $n-k$. In $n=3$ this associates an antisymmetric tensor of rank $2$ with a "vector". These are the pseudo-vectors.

This explains why the "vector product" only exists in $\mathbb{R}^3$.