Chen inequalities for immersions in trans-Sasakian space forms with slant factor

In this research article, our focus is directed towards the exploration of trans-Sasakian manifolds that incorporate a distinctive type of non-metric connection referred to as a quarter-symmetric non-metric ($QSNM$) connection. We delve into the derivation of the mathematical expressions governing the curvature tensor $\widetilde{R}$ of trans-Sasakian space forms, utilizing the aforementioned $QSNM$-connection. Our primary efforts are centered around the establishment of Chen inequalities. These inequalities find application in the characterization of slant submanifolds in the trans-Sasakian space forms and connected by a $QSNM$-connection. Furthermore, our investigation encompasses the classification of Chen invariants. This classification is extended to $\alpha$-Sasakian, $\beta$-Kenmotsu and cosymplectic manifolds, all of which are endowed with the distinctive $QSNM$-connection.