Abstract
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.
Highlights
Received: 24 January 2022One of the most fundamental problems in submanifold theory is the immersibility of a Riemannian manifold into a Euclidean space
In order to establish such relationships, the first author introduced in the 1990s the notion of δ-invariants for Riemannian manifolds
The earlier results on δ-invariants and applications have been summarized in the first author’s book [7] published in 2011. The purpose of this survey article is to present a comprehensive account of the development on the differential geometry of submanifolds in complex space forms involving δ-invariants done mostly after the publication of the book [7]
Summary
One of the most fundamental problems in submanifold theory is the immersibility of a Riemannian manifold into a Euclidean space. Nash’s embedding theorem was aimed for in the hope that if a Riemannian manifold could be regarded as isometrically embedded submanifold, this would yield the opportunity to use help from extrinsic geometry. This hope was not materialized until the publication of M. Submanifolds of higher codimension are very difficult to be understood Another reason is that at that time there did not exist general optimal relationships between the known intrinsic invariants and the main extrinsic invariants for arbitrary submanifolds of Euclidean spaces except the. The purpose of this survey article is to present a comprehensive account of the development on the differential geometry of submanifolds in complex space forms involving δ-invariants done mostly after the publication of the book [7]
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