Hodge duality

Redirect to:

• Geometry

Suppose that we have an oriented inner product space $V$ of dimension $n$. For each integer $k$ with $0 ≤ k ≤ n$, the Hodge star operator establishes a one-to-one mapping from the space of $k$-vectors $\Lambda ^k$ in $V$ and the space of $(n−k)$-vectors $\Lambda ^{n-k}$. The image of a $k$-vector $a$ under this mapping is called the Hodge dual of $a$.

Notation: $$\star : \Lambda ^k (V)\rightarrow \Lambda ^{n-k}(V).$$

The construction is based on the idea of orthogonal complement.

Applied in Hodge duality of differential forms.