Isospectral nearly K\"ahler manifolds

TL;DR

This paper introduces a systematic method for constructing isospectral nearly K"ahler manifolds using almost conjugate subgroup pairs, resulting in infinite families of such manifolds with matching spectral properties.

Contribution

It presents a novel systematic construction of isospectral nearly K"ahler manifolds via almost conjugate subgroups of specific spin groups, expanding the known examples in differential geometry.

Findings

Constructed infinite families of isospectral nearly K"ahler manifolds.

Computed volumes of six-dimensional nearly K"ahler manifolds.

Investigated Sunada pairs in six dimensions.

Abstract

We give a systematic way to construct almost conjugate pairs of finite subgroups of $Spin(2n+1)$ and $Pin(n)$ for $n\in \mathbb{N}$ sufficiently large. As a geometric application, we give an infinite family of pairs $M_1^{d_n}$ and $M_2^{d_n}$ of nearly K\"ahler manifolds that are isospectral for the Dirac and Laplace operator with increasing dimensions $d_n>6$. We provide additionally a computation of the volume of (locally) homogeneous six dimensional nearly K\"ahler manifolds and investigate the existence of Sunada pairs in this dimension.