A Riemannian invariant and its applications to submanifold theory
type: Item
instance of: scholarly article
title: A Riemannian invariant and its applications to submanifold theory
zbMATH Open document ID: 0834.53045
DOI: 10.1007/BF03322265
published in: Results in Mathematics
publication date: 1996-03-18T00:00:00Z
review text: Let \(M^n\) be a Riemannian manifold with scalar curvature \(r\). If \(K (\pi)\) denotes the sectional curvature of the 2-plane \(\pi\), then \(\delta_M(p) = r(p) - \inf K(p)\) (where the infimum is taken over all the 2-planes in \(p \in M)\) defines a Riemannian invariant [introduced by the author in ``A Riemannian invariant for submanifolds in space forms and its applications'', Geometry and topology of submanifolds, VI (World Scientific), 58-81 (1994; Zbl 0865.53050)]. Here the author uses \(\delta_M\) to derive necessary conditions for the existence of slant immersions. E.g., it is proved: Let \(M\) be compact, with finite \(\pi_1\). i) If \(\delta_M > 0\), \(M\) admits no slant immersion in any flat Kähler manifold; ii) if \(2 \delta_M > (n - 2) (n + 1)c\), \(M\) admits no Lagrangian immersion in \(\mathbb{C} P^n (4c)\). The paper is rich in examples and ends with a list of open problems in the theory of slant immersions.
reviewed by: Liviu Ornea
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 53C40
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 785192
zbMATH Keywords: minimal immersion
zbMATH Keywords: slant immersions
zbMATH Keywords: Lagrangian immersion
Wikidata QID: Q115390794
author: Bang-Yen Chen
MaRDI profile type: Publication
cites work: Calabi curves as holomorphic Legendre curves and Chen's inequality
cites work: Lagrangian surfaces in the complex Euclidean plane with conformal Maslov form
cites work: Q3999447
cites work: Some pinching and classification theorems for minimal submanifolds
cites work: Q4867770
cites work: Q4323757
cites work: Jacobi's elliptic functions and Lagrangian immersions
cites work: Totally real submanifolds of \(\mathbb{C} P^ n\) satisfying a basic equality
cites work: Characterizing a class of totally real submanifolds of \(S^ 6\) by their sectional curvatures
cites work: Q4845119
cites work: Q4012538
cites work: On Totally Real Submanifolds
cites work: Slant submanifolds in complex Euclidean spaces
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cites work: Q3918700
cites work: Q5597399
cites work: Q4311147
cites work: Construction of slant immersions. II
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description: scientific article; zbMATH DE number 785192