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 formula_1, functions on a manifold, vector valued functions, vector fields, or, more generally, sections of a vector bundle. In an invariant differential operator formula_2, the term "differential operator" indicates that the value formula_3 of the map depends only on formula_4 and the derivatives of formula_5 in formula_6. The word "invariant" indicates that the operator contains some symmetry. This means that there is a group formula_7 with a group action on the functions (or other objects in question) and this action is preserved by the operator:  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. Let "M" = "G"/"H" be a homogeneous space for a Lie group G and a Lie subgroup H. Every representation formula_9 gives rise to a vector bundle  Sections formula_11 can be identified with  In this form the group "G" acts on sections via  Now let "V" and "W" be two vector bundles over "M". Then a differential operator  that maps sections of "V" to sections of "W" is called invariant if  for all sections formula_16 in formula_17 and elements "g" in "G". All linear invariant differential operators on homogeneous parabolic geometries, i.e. when "G" is semi-simple and "H" is a parabolic subgroup, are given dually by homomorphisms of generalized Verma modules. Given two connections formula_18 and formula_19 and a one form formula_20, we have for some tensor formula_22. Given an equivalence class of connections formula_23, we say that an operator is invariant if the form of the operator does not change when we change from one connection in the equivalence class to another. For example, if we consider the equivalence class of all torsion free connections, then the tensor Q is symmetric in its lower indices, i.e. formula_24. Therefore we can compute where brackets denote skew symmetrization. This shows the invariance of the exterior derivative when acting on one forms. Equivalence classes of connections arise naturally in differential geometry, for example:    Given a metric   on formula_31, we can write the sphere formula_32 as the space of generators of the nil cone  In this way, the flat model of conformal geometry is the sphere formula_34 with formula_35 and P the stabilizer of a point in formula_31.