  This paper is concerned with the intrinsic geometric structures of conductive
transmission eigenfunctions. The geometric properties of interior transmission
eigenfunctions were first studied in [9]. It is shown in two scenarios that the
interior transmission eigenfunction must be locally vanishing near a corner of
the domain with an interior angle less than $\pi$. We significantly extend and
generalize those results in several aspects. First, we consider the conductive
transmission eigenfunctions which include the interior transmission
eigenfunctions as a special case. The geometric structures established for the
conductive transmission eigenfunctions in this paper include the results in [9]
as a special case. Second, the vanishing property of the conductive
transmission eigenfunctions is established for any corner as long as its
interior angle is not $\pi$ when the conductive transmission eigenfunctions
satisfy certain Herglotz functions approximation properties. That means, as
long as the corner singularity is not degenerate, the vanishing property holds
if the underlying conductive transmission eigenfunctions can be approximated by
a sequence of Herglotz functions under mild approximation rates. Third, the
regularity requirements on the interior transmission eigenfunctions in [9] are
significantly relaxed in the present study for the conductive transmission
eigenfunctions. Finally, as an interesting and practical application of the
obtained geometric results, we establish a unique recovery result for the
inverse problem associated with the transverse electromagnetic scattering by a
single far-field measurement in simultaneously determining a polygonal
conductive obstacle and its surface conductive parameter.
