Invariant differential operator

In mathematics and theoretical physics, an invariant differential operator is a kind of mathematical map from some objects to an object of similar type. These objects are typically functions on ${\displaystyle \mathbb {R} ^{n}}$, functions on a manifold, vector valued functions, vector fields, or, more generally, sections of a vector bundle.

In an invariant differential operator ${\displaystyle D}$, the term differential operator indicates that the value ${\displaystyle Df}$ of the map depends only on ${\displaystyle f(x)}$ and the derivatives of ${\displaystyle f}$ in ${\displaystyle x}$. The word invariant indicates that the operator contains some symmetry. This means that there is a group ${\displaystyle G}$ with a group action on the functions (or other objects in question) and this action is preserved by the operator:

${\displaystyle D(g\cdot f)=g\cdot (Df).}$

Usually, the action of the group has the meaning of a change of coordinates (change of observer) and the invariance means that the operator has the same expression in all admissible coordinates.