A fluent explanation for symmetry and particularly the dimensional reduction of electromagnetic BVPs requires clear separation of different aspects such as metric and dimension. The structures of differential geometry meet the requirements. This section gives a brief introduction to some of the essential structures we use in this paper. References with precise definitions and more detailed expositions include [2–6].

*Differentiable manifolds* serve as the domains of BVPs. For each *m*-manifold *M*, there exists a class of homeomorphisms *U* ⊂ *M* → ℝ^{
m
} called *charts*. Each point *x* of *M* has a *tangent space T*_{
x
}(*M*), which is an *m*-dimensional vector space of *tangent vectors*. Any differentiable mapping *f* : *M* → *N* between manifolds induces a unique mapping *f*_{
*
}, the *pushforward* of *f*, that maps linearly from *T*_{
x
}(*M*) to *T*_{f(x)}(*N*). An *oriented manifold* is a manifold whose tangent spaces are oriented and a *manifold-with-boundary M* is a manifold that has a boundary ∂*M* ⊂ *M*. A *diffeomorphism* is a differentiable bijection between manifolds with a differentiable inverse. An *embedded submanifold* of *M* is a pair (*N*, *f*), where *N* is a manifold and *f* : *N* → *f*(*N*) ⊂ *M* is a diffeomorphism.

Dimensional reduction is based on "smooth symmetries." This smoothness is reflected in the manifold structure of the symmetry groups. A *Lie group* is a group that is also a manifold such that the group operations are differentiable mappings. An example of a Lie group is ℝ = (ℝ, +, *id*), where ℝ is the set of real numbers, the group operation + is the addition of real numbers, and the chart *id* is the identity mapping of ℝ. Another example is *S*^{1}, the group of all complex numbers with modulus one under multiplication. ℝ and *S*^{1} are the only *connected* 1-D Lie groups up to isomorphism, and every connected 2-D Lie group is a product of them.

It is possible to define an analysis on manifolds in a coordinate- and metric-free manner and independent of the dimension. This analysis employs *differential forms*: A differential *p*-form *ω*, or a *p*-form for short, assigns each point *x* ∈ *M* an antisymmetric *p*-linear mapping *ω*_{
x
} from the tangent space *T*_{
x
}(*M*) to the real numbers. The vector space of all *p*-forms on *M* is denoted by Ω^{
p
}(*M*), and the set of all differential forms on *M* is denoted by Ω(*M*). For each differentiable mapping *f* : *M* → *N* between manifolds, there is an induced mapping *f** : Ω(*N*) → Ω(*M*) called the *pullback*. It is defined point-wise as follows: (*f***ω*)*x*(*v*_{1},..., *v*_{
p
}) = *ω*_{f(x)}(*f*_{*}*v*_{1},..., *f*_{*}*v*_{
p
}) holds for all *x* ∈ *M*, *v*_{1},..., *v*_{
p
} ∈ *T*_{
x
}(*M*). The restriction of differential forms to a submanifold *A* of *M* is given by the *trace t*_{
A
}, which is the pullback
of the inclusion map *i*_{
A
} : *A* → *M*.

The *contraction i*_{
X
} by a *vector field X* decreases the degree of a form *ω* by one such that (*i*_{
X
}*ω*)_{
x
}(*v*_{2},..., *v*_{
p
}) = *ω*_{
x
}(*X*_{
x
}, *v*_{2},..., *v*_{
p
}) holds for all *x* ∈ *M*, *v*_{2},..., *v*_{
p
} ∈ *T*_{
x
}(*M*). The *wedge product* is a bilinear mapping Λ : Ω(*M*) × Ω(*M*) → Ω(*M*) which is anticommutative. The *extension I*_{
α
} by a 1-form *α* and the wedge product increases the degree of *ω* by one: *I*_{
α
}*ω* = *α* Λ *ω*.

The differential operators *grad*, *curl*, and *div* of the vector analysis are metric counterparts of a single metric-free differential operator on differential forms called the *exterior derivative*. On a manifold *M*, it is the linear mapping *d*_{
M
} : Ω^{
p
}(*M*) → Ω^{p+1}(*M*) such that it is the *differential* for 0-forms and *d*_{
M
} (*d*_{
M
}*ω*) = 0 holds for all *ω*. Furthermore, *d* commutes with pullback: *f** ∘ *d*_{
N
} = *d*_{
M
} ∘ *f** holds for *f* : *M* → *N*.

When (ℝ, +) acts on manifold *M* such that the symmetry transformations are diffeomorphisms, then the symmetry group is called a *1-parameter group of transformations*. The action induces a smooth vector field *X* on *M* such that the vector field is everywhere tangent to the orbits, most of which are now 1-D submanifolds. With a 1-parameter group of transformations and the pullback, one can define a directional derivative of forms in the direction of the orbits. This derivative is called the *Lie derivative*, and it is denoted by
.

To assure uniqueness of a BVP solution, some *cohomology classes* of the fields may have to be specified explicitly, and by *de Rham's theorem*, this can be done by fixing the values of integrals of the fields over suitable submanifolds [7]: These integrals are presented as a linear operator
that operates on fields. Thus, the *cohomology condition* of a form *ω* is given by a real number tuple
that contains the values of the integrals.

A *metric tensor* allows a definition of *Hodge-operator* which can be used to express constitutive equations [3]. The Hodge-operator is a linear isomorphism ⋆: Ω^{
p
}(*M*) → Ω^{
m-p
}(*M*) such that it is *definite*. To give preference to the physics modeled by the constitutive equations over the metric chosen for distance measurements and modeling, we make the following generalization [6]:

**Definition 2**. *Let M be an oriented m-manifold. A definite linear isomorphism υ* : Ω^{
p
}(*M*) → Ω^{
m-p
}(*M*) *is a* Hodge-like operator *if there exists a metric tensor* *ϕ* *of M and a linear isomorphism υ*_{
ϕ
}: Ω^{m-p}(*M*) → Ω^{
m-p
}(*M*) *such that υ* = *υ*_{
ϕ
}∘⋆_{
ϕ
}*holds, where* ⋆_{
ϕ
}*is the Hodge-operator induced by* *ϕ*.