@@ -246,6 +246,14 @@ In contrast to higher dimensions, on surfaces we have density of smooth function

Since the winding number is an integer and depends continuously on the function, approximations that are close enough in the norm sense have the same winding number and are thus orientable on loops as well.

At this point the orientability criterion for continuous line fields is used and the convergence in norm gives the orientation of the approximated line field.\todo{Probably shorten this paragraph.}

The two results about orientability answer the second question \enquote{Is a given line field orientable?} in special cases.

As the next step \parencite{q4-Ball2011} presents a condition on a flat two-dimensional domain with holes for orientability of the energy minimizer of the harmonic energy $E(Q)= ∫ \abs{∇Q}\D x$.

The harmonic energy is a special case of the Frank-Oseen energy.

Unfortunately this result relies heavily on another work \parencite{q4-q6-vortices-Bethuel1994-hm} that reduces the minimization problem to a scalar problem and shows that a minimizer exists.

This work specialises on two-dimensional domains with holes and extending it is far beyond the scope of this thesis.

In \cref{sec:contraexampleTorus} we show though that there exist a condition on the torus similar to a boundary condition for which the line field minimizer of the harmonic energy is not orientable.

Therefore the question of orientability of harmonic energy minimizers on surfaces remains open.

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The mentioned density results are exploited in \cref{sec:orientability_of_sobolev_line_fields}.