Jae Won Lee

A warped product submanifold, whose tangent bundle can be decomposed to two orthogonal distributions; invariant and pointwise slant, is called a warped product pointwise semi-slant submanifold. The objective of this paper is to classify warped product pointwise semi-slant submanifolds, isometrically immersed into a Sasakian manifold. Park [25] provided the (non)-existence of a warped product pointwise semi-slant submanifold in a Sasakian manifold such that the structure vector field is tangential to fibers. In contrast, we provide intriguing theorems on warped product pointwise semi-slant submanifolds in a Sasakian manifold in terms of the shape operator and tensor fields such that the structure vector field is tangential to a base manifold and each fiber is a pointwise slant submanifold.

In this paper, we study C1 Hermite interpolation with a spatial rational Pythagorean hodograph (RPH) curve for a regular C1 Hermite data set. In particular, we completely classify simple PH M?bius cubics and prove that there exist two scaled Enneper surfaces as a PH-preserving mapping satisfying the given C H1ermite data set. Also, we give the algorithm to construct RPH curves on the Enneper surface by using PH M?bius cubics and scaled PH-preserving mappings. Finally, we calculate arc-length and bending energy of these RPH curves to choice the best RPH curves on the Enneper surface, and give some examples for RPH curves.

For surfaces in a Euclidean 3-space Casorati [4] introduced a new curvature in 1890 what is today called the Casorati curvature. This curvature was preferred by Casorati over Gauss curvature because Gauss curvature may vanish for surfaces that look intuitively curved, while Casorati curvature only vanishes at the planer points. The Casorati curvature C of submanifolds in a Riemannian manifold is the extrinsic invariant given by the normalized square of the second fundamental form and some optimal inequalities containing Casorati curvatures were obtained for submanifolds of real space forms, complex space forms, and quaternionic space forms [6,11,15,16,17,24,29]. The notion of Casorati curvature is the extended version of the notion of principal curvatures of a hypersurface of a Riemannian manifold. So, it is both important and very interesting to obtain some optimal inequalities for the Casorati curvatures of submanifolds in any ambient Rimannian manifolds. Later, C.W., J.W., Şahin and Vilcu [13] were obtained inequalities for Riemannian maps to space forms, as well as for Riemannian submersion to space forms, involving Casorati curvature. In this study, for a submersion between a space form and Riemannian manifold, we establish an optimization involving the Casorati curvature of the horizontal space. We also investigate the harmonicity of Riemannian map involving Casorati inequalities.

In this paper, we obtain a generalized Wintgen-type inequality for submanifolds in generalized (κ,µ)-space forms. Further, we discuss the inequality for bi-slant submanifolds, semi-slant submanifolds, hemi-slant submanifolds, CR-submanifolds and slant submanifolds in various ambient spaces. In particular, we generalize some recent results of Mihai, Tohoku Math. J. 69 (2017), and Aquib et al., Tamkang J. Math. 50 (2019) and Mathematics 7(12) (2019).