Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 017, 29 pages      arXiv:2007.07491      https://doi.org/10.3842/SIGMA.2021.017
Contribution to the Special Issue on Scalar and Ricci Curvature in honor of Misha Gromov on his 75th Birthday

Convergence to the Product of the Standard Spheres and Eigenvalues of the Laplacian

Masayuki Aino
RIKEN, Center for Advanced Intelligence Project AIP, 1-4-1 Nihonbashi, Tokyo 103-0027, Japan

Received July 17, 2020, in final form February 07, 2021; Published online February 24, 2021

Abstract
We show a Gromov-Hausdorff approximation to the product of the standard spheres $S^{n-p}\times S^p$ for Riemannian manifolds with positive Ricci curvature under some pinching condition on the eigenvalues of the Laplacian acting on functions and forms.

Key words: Gromov-Hausdorff distance; Lichnerowicz-Obata estimate; parallel $p$-form.

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