[Paper Review] $δ$-Invariants, Inequalities of Submanifolds and Their Applications

Bang‐Yen Chen|arXiv (Cornell University)|Jul 7, 2013

Geometric Analysis and Curvature FlowsMathematics139 references22 citations

TL;DR

This paper introduces $\delta$-invariants—novel Riemannian invariants distinct from classical scalar and Ricci curvatures—to establish optimal inequalities relating intrinsic and extrinsic geometry of submanifolds. It demonstrates that these invariants resolve long-standing obstructions in minimal immersions and provide powerful tools for estimating eigenvalues, classifying ideal immersions, and analyzing submanifolds in complex, contact, and Einstein manifolds.

ABSTRACT

The famous Nash embedding theorem was aimed for in the hope that if Riemannian manifolds could be regarded as Riemannian submanifolds, this would then yield the opportunity to use extrinsic help. However, as late as 1985 (see \cite{G}) this hope had not been materialized. The main reason for this is due to the lack of controls of the extrinsic properties of the submanifolds by the known intrinsic invariants. In order to overcome such difficulties as well as to provide answers to an open question on minimal immersions, we introduced in the early 1990's new types of Riemannian invariants, known as the $δ$-invariants or the so-called Chen invariants, different in nature from the "classical" Ricci and scalar curvatures. At the same time we also able to establish general optimal relations between the new intrinsic invariants and the main extrinsic invariants for Riemannian submanifolds. Since then many results concerning these invariants, inequalities, related subjects, and their applications have been obtained by many geometers. The main purpose of this article is thus to provide an extensive and comprehensive survey of results over this very active field of research done during the last fifteen years. Several related inequalities and their applications are presented in this survey article as well.

Motivation & Objective

• To address the lack of control over extrinsic invariants (e.g., mean curvature) by classical intrinsic invariants (e.g., Ricci curvature) in submanifold theory.
• To overcome the limitations of Nash's embedding theorem in practical geometric analysis by introducing new intrinsic invariants that better reflect geometric constraints.
• To solve Chern's open problem on necessary conditions for minimal isometric immersions into Euclidean spaces by introducing $\delta$-invariants.
• To unify and generalize existing curvature inequalities and provide optimal relations between intrinsic and extrinsic invariants across diverse geometric settings.
• To extend the applicability of curvature invariants to complex, contact, Lagrangian, and affine differential geometry, and to general relativity.

Proposed method

• Define $\delta$-invariants as scalar curvature-based invariants that subtract specific sectional curvature terms, yielding new intrinsic curvature invariants sensitive to submanifold geometry.
• Establish general optimal inequalities linking $\delta$-invariants with extrinsic invariants such as mean curvature $H$, scalar curvature $\tau$, and Ricci curvature.
• Use the concept of 'ideal immersions'—submanifolds achieving equality in $\delta$-inequalities—to characterize optimal geometric configurations.
• Apply the $\delta$-invariant framework to derive eigenvalue estimates for the Laplacian on compact submanifolds via curvature bounds.
• Extend results to specialized settings: CR-submanifolds, Lagrangian and slant submanifolds, warped products, and Riemannian submersions.
• Utilize the DDVV conjecture and related inequalities to analyze normal scalar curvature and submanifold rigidity.

Experimental results

Research questions

• RQ1 Can new intrinsic invariants be defined to control extrinsic geometry of submanifolds more effectively than classical curvature invariants?
• RQ2 What are the optimal inequalities relating $\delta$-invariants to mean curvature and other extrinsic invariants in submanifold geometry?
• RQ3 How can $\delta$-invariants be used to characterize minimal or totally real immersions in complex space forms?
• RQ4 What are the implications of $\delta$-invariant inequalities for the spectrum of the Laplacian on compact submanifolds?
• RQ5 In what ways do $\delta$-invariants enhance rigidity results and classification theorems for submanifolds in Einstein, conformally flat, and contact manifolds?

Key findings

• The $\delta$-invariants provide a new class of intrinsic curvature invariants that are fundamentally different from scalar and Ricci curvatures, as they are derived by subtracting specific sectional curvature terms from the total scalar curvature.
• Optimal inequalities are established between $\delta$-invariants and extrinsic invariants such as mean curvature $H$, with equality characterizing 'ideal immersions' that minimize the normalized scalar curvature of the ambient space.
• For purely real submanifolds in complex space forms $\tilde{M}^n(4\epsilon)$, the inequality $H^2 \geq \frac{2(n+2)}{n^2(n-1)}\tau - \frac{n+2}{n}\left[1 + \frac{3\|P\|^2}{n(n-1)}\right]\epsilon$ holds, with equality under specific geometric conditions.
• The $\delta(2)$-invariant is shown to be particularly effective in classifying Lagrangian and $CR$-submanifolds, especially in nearly Kähler $S^6$.
• Applications to warped products yield growth estimates for warping functions, and the $\delta$-invariant framework enables new rigidity results in submanifold theory.
• The theory provides a systematic approach to estimating eigenvalues of the Laplacian on compact submanifolds via curvature bounds derived from $\delta$-invariants.