An Extension of Gauss-Bonnet's Theorem C: Gauss-Bonnet Theorem in Non-Hermitian Systems: Skin Effect and Topological Conservation

This paper investigates the extension of the Gauss-Bonnet theorem in non
Hermitian systems. By introducing the concepts of non-Hermitian curvature and
 boundary localization phenomena induced by the skin effect, we establish a gen
eralized Gauss-Bonnet formula applicable to non-reciprocal systems. This formula
 maintains topological conservation while quantifying the impact of boundary skin
 effects on geometric-topological relationships through a jump index. The the
oretical framework is mathematically rigorous and self-consistent, with physical
 applications in non-Hermitian topological photonic crystals and circuit systems,
 providing new theoretical tools for understanding topological phase transitions in
 non-reciprocal systems.