General Chen's first inequality and applications for Riemannian maps
Ravindra Singh, Kiran Meena, Kapish Chand Meena
TL;DR
This work develops a universal Chen's first inequality for Riemannian maps between Riemannian manifolds, linking horizontal and range curvatures through the map's tension field and second fundamental form via a sharp lower bound on K^H(P). It then specializes the general inequality to targets that are generalized complex and generalized Sasakian space forms, yielding explicit curvature bounds in terms of structure functions f1,f2,f3 and decomposition data such as P, Θ(P), and Φ(P). The authors extract corollaries for classical space forms (real, complex, real Kähler, Sasakian, Kenmotsu, cosymplectic, almost C(α)) and provide δ-invariant estimations, including simplifications under harmonic map conditions. Overall, the paper offers a unified framework to analyze curvature relations for Riemannian maps into a broad class of space forms, with potential implications for rigidity, geometric flows, and intrinsic–extrinsic comparison results.
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 Kähler, Sasakian, Kenmotsu, cosymplectic, and almost $C(α)$ space forms. Moreover, we also discuss geometric applications of CFI on generalized complex and generalized Sasakian space forms. Specifically, we give estimations for $δ$-invariants within various hypotheses.