Recent developments in δ-Casorati curvature invariants

Chen, Bang‐Yen

doi:10.3906/mat-2009-8

Cited by 18 publications

(11 citation statements)

References 53 publications

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“…For further results on δ-Casorati curvatures, we refer to the recent survey [100] on δ-Casorati curvature invariants and to those articles cited in [100]. We also note that the counterpart of Theorems 55, 56 and 59 in Sasakian geometry can be find in the recent articles [101,102], while the extension of the above theorems in quaternion setting is still an open problem.…”

Section: Remark 13mentioning

confidence: 98%

Differential Geometry of Submanifolds in Complex Space Forms Involving δ-Invariants

2022

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One of the fundamental problems in the theory of submanifolds is to establish optimal relationships between intrinsic and extrinsic invariants for submanifolds. In order to establish such relations, the first author introduced in the 1990s the notion of δ-invariants for Riemannian manifolds, which are different in nature from the classical curvature invariants. The earlier results on δ-invariants and their applications have been summarized in the first author’s book published in 2011 Pseudo-Riemannian Geometry, δ-Invariants and Applications (ISBN: 978-981-4329-63-7). In this survey, we present a comprehensive account of the development of the differential geometry of submanifolds in complex space forms involving the δ-invariants done mostly after the publication of the book.

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Section: Remark 13mentioning

confidence: 98%

Differential Geometry of Submanifolds in Complex Space Forms Involving δ-Invariants

2022

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“…Similar to the definition of a Riemannian manifold of quasi-constant curvature given in [13], we define a statistical manifold of quasi-constant curvature as follows: (10) where α, β are scalar functions and η is a 1-form defined by…”

Section: Statistical Manifolds Of Qc Curvature and Their Submanifoldsmentioning

confidence: 99%

Optimal inequalities for submanifolds in statistical manifolds of quasi constant curvature

Bansal

Uddin

Shahid

2021

Filomat

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In this paper, we establish B.-Y. Chen?s optimal inequalities for statistical submanifolds involving Casorati curvature and the normalized scalar curavture in a statistical manifold of quasi constant curvature. The equality cases of these inequalities are also considered. Further, we provide some applications of our results. Moreover, as a new example we construct minimal statistical surface (statistical submanifold) of a statistical manifold of quasi constant curvature.

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“…Very recently, the authors in [4] studied the information-geometric properties of the statistical manifold to reduce predictive uncertainly via data assimilation. On the other hand, statistical manifolds provide a setting for the theory of submanifolds, where a basic problem is to find out simple relationships between the main intrinsic and extrinsic curvature invariants of submanifolds [5]. In this regard, many geometers studied certain types of geometric inequalities in (statistical) submanifolds.…”

Section: Introductionmentioning

confidence: 99%

“…In this regard, many geometers studied certain types of geometric inequalities in (statistical) submanifolds. Comprehensive surveys on such inequalities are provided by Chen in [5][6][7].…”

Section: Introductionmentioning

confidence: 99%

“…Initially, the δ-Casorati curvatures were studied in optimal inequalities for submanifolds in real space forms in [8,9]. Later, this topic was extensively studied (e.g., see [5]). Recently, Suceavȃ and Vajiac established inequalities involving some Chen invariants, mean curvature, and Casorati curvature for strictly convex Euclidean hypersurfaces [15].…”

Section: Introductionmentioning

confidence: 99%

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Inequalities for the Casorati Curvature of Totally Real Spacelike Submanifolds in Statistical Manifolds of Type Para-Kähler Space Forms

Chen

Decu

Vîlcu

2021

Entropy

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Recent developments in δ-Casorati curvature invariants

Cited by 18 publications

References 53 publications

Differential Geometry of Submanifolds in Complex Space Forms Involving δ-Invariants

Differential Geometry of Submanifolds in Complex Space Forms Involving δ-Invariants

Optimal inequalities for submanifolds in statistical manifolds of quasi constant curvature

Inequalities for the Casorati Curvature of Totally Real Spacelike Submanifolds in Statistical Manifolds of Type Para-Kähler Space Forms

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