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  Volume 6, Issue 3, Article 77
A General Optimal Inequality for Arbitrary Riemannian Submanifolds

    Authors: Bang-Yen Chen,  
    Keywords: $delta$-invariants, Inequality, Riemannian submanifold, Squared mean curvature, Sectional curvature.  
    Date Received: 22/03/05  
    Date Accepted: 28/07/05  
    Subject Codes:

53C40, 53C42, 53B25

    Editors: Wing-Sum Cheung,  

One of the most fundamental problems in submanifold theory is to establish simple relationships between intrinsic and extrinsic invariants of the submanifolds (cf. [6]). A general optimal inequality for submanifolds in Riemannian manifolds of constant sectional curvature was obtained in an earlier article [5]. In this article we extend this inequality to a general optimal inequality for arbitrary Riemannian submanifolds in an arbitrary Riemannian manifold. This new inequality involves only the $ delta$ -invariants, the squared mean curvature of the submanifolds and the maximum sectional curvature of the ambient manifold. Several applications of this new general inequality are also presented.

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