The geometric description of D-branes in WZW models is pushed forward. Our
starting point is a gluing condition\, J+=FJ− that matches the model's
chiral currents at the worldsheet boundary through a linear map F acting on
the WZW Lie algebra. The equivalence of boundary and gluing conditions of this
type is studied in detail. The analysis involves a thorough discussion of
Frobenius integrability, shows that F must be an isometry, and applies to
both metrically degenerate and nondegenerate D-branes. The isometry F need
not be a Lie algebra automorphism nor constantly defined over the brane. This
approach, when applied to isometries of the form F=R with R a constant Lie
algebra automorphism, validates metrically degenerate R-twined conjugacy
classes as D-branes. It also shows that no D-branes exist in semisimple WZW
models for constant\, F=−R.Comment: 23 pages, discussion of limitations of the gluing condition approach
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