Ion Mihai

The sectional curvature, Ricci curvature, and scalar curvature for a product generalized Sasakian space form are obtained. Furthermore, the Chen-Ricci inequality and the Hineva inequality are established for submanifolds of a product generalized Sasakian space form, including product Sasakian, product cosymplectic, and product Kenmotsu space forms. The equality cases are also discussed.

The study of complex space forms plays a central role in differential geometry, as these manifolds provide a natural framework for exploring geometric structures endowed with rich symmetries, extending both Riemannian and Kählerian geometries. In this paper, we extend the definition of Kählerian slant submanifolds when the ambient complex space form admits a quarter-symmetric connection. Furthermore, we establish sharp relationships between intrinsic and extrinsic geometric invariants of Kählerian slant submanifolds in complex space forms admitting a quarter-symmetric connection.

In the present paper, we give a simple proof of the Chen–Ricci inequality for submanifolds in Riemannian and Lorentzian space forms, respectively. Moreover, we extend the Chen–Ricci inequality to submanifolds in Lorentzian manifolds with a semi-symmetric non-metric connection.

The current work establishes the geometrical bearing for hypersurfaces in a Golden Riemannian manifold with constant golden sectional curvature with respect to k-almost Newton-conformal Ricci solitons. Moreover, we extensively explore the immersed r-almost Newton-conformal Ricci soliton and determine the sufficient conditions for total geodesicity with adequate restrictions on some smooth functions using mathematical operators. Furthermore, we go over some natural conclusions in which the gradient k-almost Newton-conformal Ricci soliton on the hypersurface of the Golden Riemannian manifold becomes compact. Finally, we establish a Schur’s type inequality in terms of k-almost Newton-conformal Ricci solitons immersed in Golden Riemannian manifolds with constant golden sectional curvature.

We establish an improved Chen inequality involving scalar curvature and mean curvature and geometric inequalities for Casorati curvatures, on slant submanifolds in a Lorentzian–Sasakian space form endowed with a semi-symmetric non-metric connection. Also, we present examples of slant submanifolds in a Lorentzian–Sasakian space form.

The geometry of submanifolds in Kähler manifolds is an important research topic. In the present paper, we study submanifolds in complex space forms admitting a semi-symmetric non-metric connection. We prove the Chen–Ricci inequality, Chen basic inequality, and a generalized Euler inequality for such submanifolds. These inequalities provide estimations of the mean curvature (the main extrinsic invariants) in terms of intrinsic invariants: Ricci curvature, the Chen invariant, and scalar curvature. In the proofs, we use the sectional curvature of a semi-symmetric, non-metric connection recently defined by A. Mihai and the first author, as well as its properties.