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Geometric and topological rigidity of pinched submanifolds in Riemannian manifolds

Theodoros Vlachos

Abstract

We study the geometry and topology of compact submanifolds of Riemannian manifolds with arbitrary codimension that satisfy a pinching condition. The ambient space is not required to be a space form, and the pinching condition involves the length of the second fundamental form and the mean curvature. The results obtained, as well as the methods employed, differ substantially between higher-dimensional and four-dimensional submanifolds. Our approach in higher dimensions relies on results from the theory of Riemannian manifolds with nonnegative isotropic curvature and on the Bochner technique. In contrast, in the four-dimensional case, the analysis also relies critically on concepts from four-dimensional geometry. A key observation is that submanifolds satisfying the pinching condition necessarily have nonnegative isotropic curvature, a notion introduced by Micallef and Moore. The results are sharp and extend previous work by several authors, without imposing any additional assumptions on either the topology or the geometry of the submanifold.

Geometric and topological rigidity of pinched submanifolds in Riemannian manifolds

Abstract

We study the geometry and topology of compact submanifolds of Riemannian manifolds with arbitrary codimension that satisfy a pinching condition. The ambient space is not required to be a space form, and the pinching condition involves the length of the second fundamental form and the mean curvature. The results obtained, as well as the methods employed, differ substantially between higher-dimensional and four-dimensional submanifolds. Our approach in higher dimensions relies on results from the theory of Riemannian manifolds with nonnegative isotropic curvature and on the Bochner technique. In contrast, in the four-dimensional case, the analysis also relies critically on concepts from four-dimensional geometry. A key observation is that submanifolds satisfying the pinching condition necessarily have nonnegative isotropic curvature, a notion introduced by Micallef and Moore. The results are sharp and extend previous work by several authors, without imposing any additional assumptions on either the topology or the geometry of the submanifold.
Paper Structure (19 sections, 26 theorems, 245 equations)

This paper contains 19 sections, 26 theorems, 245 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and auxiliary results
  3. Isotropic curvature
  4. The pinching condition and the isotropic curvature
  5. Isometric immersions of Riemannian products
  6. The Bochner-Weitzenböck operator
  7. Even dimensional submanifolds
  8. Proof of the result for $n\geq5$
  9. Submanifolds with finite fundamental group
  10. Submanifolds with infinite fundamental group
  11. Proof of the result for four-submanifolds
  12. Geometry of 4-dimensional manifolds
  13. Auxiliary results for 4-dimensional submanifolds
  14. Four-dimensional submanifolds with finite fundamental group
  15. Four-dimensional submanifolds with infinite fundamental group
  16. ...and 4 more sections

Key Result

Theorem 1

Let $f\colon M^n\to\accentset{\sim}{M}^{n+m}$, $n\geq 5$, be an isometric immersion of a compact Riemannian manifold. Assume that inequality 1 is satisfied and that $\accentset{\sim}{K}^f_{\min}\geq0$ at every point of $M^n$. Then one of the following holds: $(i)$ The manifold $M^n$ is diffeomorphic In particular, if $n$ is even and the second Betti number satisfies $\beta_2(M^n)>0$, then $f$ is t

Theorems & Definitions (35)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Corollary 5
  • Corollary 6
  • Theorem 7
  • Theorem 8
  • Lemma 9
  • Lemma 10
  • ...and 25 more