Integrability of the Clifford-induced almost complex structure on level hypersurfaces of OT-FKM isoparametric families

Tang and Yan (2022) constructed a natural almost complex structure $J$ on the middle level hypersurface $M_0=F^{-1}(0)\cap S^{2l-1}$ of every isoparametric family of OT-FKM type in a unit sphere, defined directly from a symmetric Clifford system $\{P_0,\dots,P_m\}$, and showed $(M_0,J)$ is almost Hermitian while leaving integrability open. A subsequent paper of Qian, Tang and Yan proved that for $m=1$ this $J$ is not integrable, and constructed different integrable structures on the $m=1$ hypersurfaces and complex structures on focal submanifolds for $m=2$ and $m=4$ (definite case). The integrability of the original Clifford-induced $J$ for Clifford systems with $m\ge 2$ remains undetermined. The problem asks to decide, for each $m\ge 2$ and each family type, whether the Nijenhuis tensor of $J$ vanishes identically, i.e. whether $(M_0,J)$ is a complex manifold.

Background

A hypersurface of the unit sphere is called isoparametric if it has constant principal curvatures. Work of É. Cartan and H.-F. Münzner shows that the number $g$ of distinct principal curvatures of such a hypersurface can only be $1,2,3,4$ or $6$, and the classification of all isoparametric hypersurfaces in unit spheres was completed around 2020. The case $g=4$ carries the richest topology, and apart from two homogeneous families all $g=4$ isoparametric hypersurfaces are of the OT-FKM type constructed by Ozeki–Takeuchi and Ferus–Karcher–Münzner, as follows. A symmetric Clifford system is a family of symmetric $2l\times 2l$ matrices $\{P_0,\dots,P_m\}$ satisfying $P_iP_j+P_jP_i=2\delta_{ij}I_{2l}$ (the defining relations of a Clifford algebra represented by symmetric matrices). From it one builds the quartic polynomial $F(x)=|x|^4-2\sum_{i=0}^m\langle P_ix,x\rangle^2$ on $\mathbb{R}^{2l}$; its restriction $f$ to the unit sphere $S^{2l-1}$ takes values in $[-1,1]$, its regular level sets $M$ are hypersurfaces with four principal curvatures of multiplicities $(m,l-m-1)$ (repeated periodically), and its singular levels $M_+=f^{-1}(1)$, $M_-=f^{-1}(-1)$ are the focal submanifolds. Here $l=k\delta(m)$ with $k$ a positive integer with $l-m-1>0$, where $\delta(m)$ is the dimension of an irreducible module of a Clifford algebra ($\delta(m)=1,2,4,4,8,8,8,8$ for $m=1,\dots,8$, then periodic up to a factor 16); when $m\equiv 0\pmod 4$ there are two non-congruent family types, distinguished by whether $P_0P_1\cdots P_m=\pm I_{2l}$ (definite) or not (indefinite).

An almost complex structure on an even-dimensional manifold is a bundle endomorphism $J:TM\to TM$ with $J^2=-\mathrm{Id}$; it is integrable when it comes from an underlying complex manifold structure. By the Newlander–Nirenberg theorem, $J$ is integrable if and only if its Nijenhuis tensor $N(X,Y)=[JX,JY]-[X,Y]-J[JX,Y]-J[X,JY]$ vanishes identically. An almost Hermitian manifold is one where $J$ is compatible with the metric. Among spheres, only $S^2$ and $S^6$ admit almost complex structures (Borel–Serre); the natural octonionic structure on $S^6$ is not integrable, and whether $S^6$ admits any complex structure is a famous open problem. Explicit compact complex manifolds are otherwise rare — the classical examples are Hopf manifolds $S^1\times S^{2n-1}$ (Peng–Tang gave a general integrability criterion for such products) and Calabi–Eckmann manifolds $S^{2p+1}\times S^{2q+1}$.

Tang and Yan (2022) constructed the following structure on the middle level hypersurface $M_0=F^{-1}(0)\cap S^{2l-1}$. The unit normal of $M_0$ at $x$ is $\xi(x)=x-2\sum_{i=0}^m\langle P_ix,x\rangle P_ix$, and the vectors $\eta_1=P_0P_1x$ and $\eta_2=P_0P_1\xi$ are tangent to $M_0$; writing $T_xM_0=\mathrm{Span}\{\eta_1\}\oplus\mathrm{Span}\{\eta_2\}\oplus E$, where $E$ denotes the orthogonal complement of $\mathrm{Span}\{\eta_1,\eta_2\}$ in $T_xM_0$ (Clifford identities show $P_0P_1$ preserves $E$), they define

$J\eta_1=\eta_2,\qquad J\eta_2=-\eta_1,\qquad JX=P_0P_1X\ \ \text{for }X\in E,$

and prove that $(M_0,J,ds^2)$ is an almost Hermitian manifold with the metric induced from $\mathbb{R}^{2l}$. Its integrability was explicitly deferred. For $m=1$ the same authors defined a different structure on $S^1\times M_+$ (all regular levels are diffeomorphic to $S^m\times M_+$) and proved it integrable by pulling back the standard complex structure of $\mathbb{R}^{2k}$; combined with the fact that among $m=1$ focal submanifolds only $M_+^5\cong S^3\times S^2$ and $M_+^{13}\cong S^7\times S^6$ are sphere products (Qian–Tang–Yan), this exhibited complex structures on $S^1\times S^3\times S^2$ (with vanishing first Chern class) and on $S^1\times S^7\times S^6$.

Problem Statement

For every symmetric Clifford system $\{P_0,\dots,P_m\}$ on $\mathbb{R}^{2l}$ with $m\ge 2$ (so $l=k\delta(m)$ with $l-m-1>0$, $\delta(m)$ being the irreducible Clifford module dimension as defined above, including both the definite and the indefinite families when $m\equiv 0\pmod 4$), consider the middle level hypersurface $M_0=F^{-1}(0)\cap S^{2l-1}$ of the OT-FKM isoparametric family, $F(x)=|x|^4-2\sum_{i=0}^m\langle P_ix,x\rangle^2$, equipped with the almost complex structure of Tang–Yan (2022): with $\xi(x)=x-2\sum_{i=0}^m\langle P_ix,x\rangle P_ix$ the unit normal, $\eta_1=P_0P_1x$, $\eta_2=P_0P_1\xi$, and $T_xM_0=\mathrm{Span}\{\eta_1\}\oplus\mathrm{Span}\{\eta_2\}\oplus E$, where $E$ is the orthogonal complement of $\mathrm{Span}\{\eta_1,\eta_2\}$ in $T_xM_0$, they set

$J\eta_1=\eta_2,\qquad J\eta_2=-\eta_1,\qquad JX=P_0P_1X\ \ \text{for }X\in E.$

Determine whether $J$ is integrable, i.e. whether its Nijenhuis tensor $N(X,Y)=[JX,JY]-[X,Y]-J[JX,Y]-J[X,JY]$ vanishes identically on $M_0$, so that $(M_0,J)$ is a complex manifold, for each $m\ge 2$ and each family type (definite/indefinite when applicable). The case $m=1$ is already settled ($J$ is not integrable) and is not part of what must be decided.

References

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