The surfaces with idempotent shape operator matrix defined along a surface curve

Abstract

In this study, we investigate surfaces in differential geometry whose shape operator is idempotent. Such an algebraic constraint on the operator imposes strict geometric restrictions on the surface, particularly on its curvature functions. We classify surfaces according to the values of their Gaussian curvature, mean curvature, and principal curvatures when the shape operator matrix defined along a surface curve satisfies $S^{2}=S$.

The results show that the idempotency condition leads to three distinct geometric cases depending on whether the geodesic torsion vanishes. These classifications provide insight into the structure of flat, minimal, elliptic, and umbilical surfaces.