Curvature Pinching Conditions in Quaternionic Manifolds Under Quarter-Symmetric Metric Connections

This work presents a pair of sharp geometric inequalities that connect the normalized scalar curvature with the generalized normalized δ-Casorati curvature for θ-slant submanifolds immersed in quaternionic space forms endowed with a quarter-symmetric metric connection (QSMC). Alongside establishing these estimates, we rigorously describe the geometric conditions under which equality is achieved. The results not only generalize prior findings related to Casorati curvature but also offer new insights into the extrinsic geometry of submanifolds under non-standard connections. To conclude, we propose several open problems that invite further exploration in this direction.