Optimal inequalities for Riemannian maps and Riemannian submersions involving Casorati curvatures
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Optimal inequalities for Riemannian maps and Riemannian submersions involving Casorati curvatures Chul Woo Lee1 · Jae Won Lee2 · Bayram Şahin3 · Gabriel‑Eduard Vîlcu4,5 Received: 27 June 2020 / Accepted: 11 September 2020 © Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Riemannian maps are generalizations of well-known notions of isometric immersions and Riemannian submersions. Most optimal inequalities on submanifolds in various ambient spaces are driven from isometric immersions. The main aim of this paper is to obtain optimal inequalities for Riemannian maps to space forms, as well as for Riemannian submersions from space forms, involving Casorati curvatures. Keywords Riemannian map · Casorati curvature · 𝛿-Casorati curvature · Normalized scalar curvature Mathematics Subject Classification 53C05 · 49K35 · 62B10
* Jae Won Lee [email protected] Chul Woo Lee [email protected] Bayram Şahin [email protected] Gabriel‑Eduard Vîlcu gvilcu@upg‑ploiesti.ro 1
Department of Mathematics, Kyungpook National University, Daegu 41566, South Korea
2
Department of Mathematics Education and RINS, Gyeongsang National University, Jinju 52828, South Korea
3
Department of Mathematics, Ege University, Izmir, Turkey
4
Research Center in Geometry, Topology and Algebra, Faculty of Mathematics and Computer Science, University of Bucharest, Str. Academiei, Nr. 14, Sector 1, 70109 Bucureşti, Romania
5
Department of Cybernetics, Economic Informatics, Finance and Accountancy, Petroleum-Gas University of Ploieşti, Bd. Bucureşti, Nr. 39, 100680 Ploieşti, Romania
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1 Introduction Casorati introduced in [5] a very natural concept of curvature for regular surfaces in the three-dimensional Euclidean space, nowadays called a Casorati curvature, as the normalized sum of the squared principal curvatures of the surface. For the general framework of manifolds, the Casorati curvature of a submanifold in a Riemannian manifold is defined as the normalized square of the length of the second fundamental form, and it is well known that this is an extrinsic invariant [9]. Such a curvature is applied to the visual perception of shapes and appearances [24]. Moreover, there are some optimal inequalities involving Casorati curvatures, proved for several submanifolds in real, complex, Sasakian, Kenmotsu and quaternionic space forms (see, e.g., [3, 4, 8, 14, 16, 18–20, 27, 35]). On the other hand, Fischer [12] introduced a broad term of isometric immersions and Riemannian submersions, called Riemannian maps between Riemannian manifolds in 1992. This notion plays an important geometric structures of two Rie( ) role in ( comparing ) mannian manifolds. Let F ∶ M m , gM ⟶ N n , gN be a smooth map between two smooth ( ) ( ) Riemannian manifolds M m , gM and N n , gN of finite dimensions m and n, respectively. If p ∈ M , then the tangent space Tp M splits orthogonally as
Tp M = Vp ⊕ Hp , ( )⟂ where Vp = kerF∗p is the vertical subspac
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