  Although the analysis of observational data indicates that quasi-separatrix
layers (QSLs) have to play an important role in magnetic configurations of
solar flares, the corresponding theory is only at an initial stage so far. In
particular, there is still a need in a proper definition of QSL. This problem
is analyzed here on the basis of geometrical properties of the mapping produced
by the field lines which connect photospheric areas of positive and negative
magnetic polarities of active regions. In general, one can find on the
photosphere a unique pair of locally perpendicular line elements at one
footpoint of a given field line, so that this pair is mapped into a similar
pair of elements at the other footpoint. Along the directions of these elements
the field line mapping only stretches and compresses the corresponding
displacements of the neighboring field-line footpoints. This fact enables us to
get a remarkably clear and concise definition of QSL as a volume filled by
coronal field lines for which the ratio of the corresponding stretching and
compressing coefficients is anomalously large. The new definition is also
compared with the ones previously introduced by other authors.
  The theory is applied to flare events of the so-called sigmoid-type, using an
analytical model of a twisted force-free configuration. It is shown that such a
configuration may contain a QSL even if genuine separatrix surfaces are absent.
Identifying QSLs is an essential prerequisite for understanding the mechanism
of magnetic energy release in this type of flares. It is also demonstrated that
the magnetic field under study has a Hamiltonian structure, which makes it
possible to reveal a geometrical reason for the appearance of QSL in this case.
