A general inequality for conformally flat submanifolds and its applications

Chen, Bang‐Yen

doi:10.1007/s10474-005-0016-2

Cited by 5 publications

(5 citation statements)

References 14 publications

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“…Conversely, if r = n(n −1) 2k+1−n holds, then it follows from (5.13) and Theorem 3.1 that M n is δ(k) -ideal. An obstruction for a manifold to be conformally flat in terms of Chen's δ -curvatures was given in [31].…”

Section: Casorati Ideal Submanifoldsmentioning

confidence: 99%

Recent developments in δ-Casorati curvature invariants

Chen¹

2021

Turk J Math

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Section: Casorati Ideal Submanifoldsmentioning

confidence: 99%

Recent developments in δ-Casorati curvature invariants

Chen¹

2021

Turk J Math

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“…Problem 7.1 Like in [14], to obtain Casorati inequalities for conformally flat submanifolds of a real space form.…”

Section: Further Studiesmentioning

confidence: 99%

Inequalities for algebraic Casorati curvatures and their applications

Tripathi

2016

Preprint

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The notion of different kind of algebraic Casorati curvatures are introduced. Some results expressing basic Casorati inequalities for algebraic Casorati curvatures are presented. Equality cases are also discussed. As a simple application, basic Casorati inequalities for different δ-Casorati curvatures for Riemannian submanifolds are presented. Further applying these results, Casorati inequalities for Riemannian submanifolds of real space forms are obtained. Finally, some problems are presented for further studies.

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“…The first author was able to establish optimal general solutions to Problem 2 in [4,5,7,8] for the families of warped product manifolds, Einstein manifolds, conformally flat manifolds, and Riemannian manifolds admitting a Riemannian submersion with totally geodesic fibers, respectively; also with Mihai in [10] for the family of Sasakian manifolds.…”

Section: Problem 2 Let F Be a Special Family Of Riemannian Manifoldsmentioning

confidence: 99%

“…Earlier, the first author proved in [7] the following optimal general inequality for the family of conformally flat submanifolds in real space forms:…”

Section: Problem 2 Let F Be a Special Family Of Riemannian Manifoldsmentioning

confidence: 99%

An extremal class of conformally flat submanifolds in Euclidean spaces

Chen

Garay

2006

Acta Math Hung

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Let M n be a Riemannian n-manifold with n = 4. Consider the Riemannian invariant σ(2) defined by σ(2) = τ − (n − 1) min Ric n 2 − 3n + 4 , where τ is the scalar curvature of M n and (min Ric)(p) is the minimum of the Ricci curvature of M n at p. In an earlier article, B. Y. Chen established the following sharp general inequality: σ(2) 5 n 2 (n − 2) 2 2(n 2 − 3n + 4) H 2 for arbitrary n-dimensional conformally flat submanifolds in a Euclidean space, where H 2 denotes the squared mean curvature. The main purpose of this paper is to completely classify the extremal class of conformally flat submanifolds which satisfy the equality case of the above inequality. Our main result states that except open portions of totally geodesic n-planes, open portions of spherical hypercylinders and open portion of round hypercones, conformally flat submanifolds satifying the equality case of the inequality are obtained from some loci of (n − 2)-spheres around some special coordinate-minimal surfaces.

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A general inequality for conformally flat submanifolds and its applications

Abstract: We establish a general sharp inequality which holds for the family of conformally flat submanifolds in real space form. We show that this inequality does not hold for arbitrary submanifolds in real space forms. We also provide several applications of the inequality.

Cited by 5 publications

References 14 publications

Recent developments in δ-Casorati curvature invariants

Recent developments in δ-Casorati curvature invariants

Inequalities for algebraic Casorati curvatures and their applications

An extremal class of conformally flat submanifolds in Euclidean spaces

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