General Chen's first inequality and applications for Riemannian maps

Abstract: In this paper, we present a general form of Chen's first inequality (in short, CFI) for Riemannian maps between Riemannian manifolds. Further, using this general form, we obtain the CFI for Riemannian maps when target spaces are generalized complex and generalized Sasakian space forms. Consequently, we also obtain the CFI for Riemannian maps when the target spaces are real, complex, real K\"ahler, Sasakian, Kenmotsu, cosymplectic, and almost $C(\alpha)$ space forms. Moreover, we also discuss geometric applications of CFI on generalized complex and generalized Sasakian space forms. Specifically, we give estimations for $\delta$-invariants within various hypotheses.