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<p>We investigate the geometry and topology of compact submanifolds of arbitrary codimension in space forms satisfying a certain pinching condition involving the length of the second fundamental form and the mean curvature. We prove that this pinching condition either forces homology to vanish in a range of intermediate dimensions, or completely determines the submanifold up to congruence. The results are sharp and extend earlier work by several authors, without imposing any additional assumptions on the mean curvature.</p>

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Geometric and topological rigidity of pinched submanifolds

  • 期刊论文

  • 发表日期：

    2025年9月23日

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  • Theodoros Vlachos

摘要

We investigate the geometry and topology of compact submanifolds of arbitrary codimension in space forms satisfying a certain pinching condition involving the length of the second fundamental form and the mean curvature. We prove that this pinching condition either forces homology to vanish in a range of intermediate dimensions, or completely determines the submanifold up to congruence. The results are sharp and extend earlier work by several authors, without imposing any additional assumptions on the mean curvature.