A new look at Condition A

TL;DR

This paper provides a geometric proof that Condition A implies an isoparametric hypersurface is of OT-FKM type, emphasizing the role of octonian algebra, and aims to clarify the classification for specific multiplicity pairs.

Contribution

It offers a straightforward, geometric proof of Dorfmeister and Neher's result, highlighting the significance of octonian algebra in the structure of isoparametric hypersurfaces.

Findings

Proof emphasizes geometric considerations over algebraic classification.

Octonian algebra underpins the structure of isoparametric hypersurfaces with Condition A.

Clarifies the role of Condition A in classifying hypersurfaces for specific multiplicity pairs.

Abstract

Ozeki and Takeuchi \cite[I]{OT} introduced the notion of Condition A and Condition B to construct two classes of inhomogeneous isoparametric hypersurfaces with four principal curvatures in spheres, which were later generalized by Ferus, Karcher and M\"unzner to many more examples via the Clifford representations; we will refer to these examples of Ozeki and Takeuchi and of Ferus, Karcher and M\"unzner collectively as OT-FKM type throughout the paper. Dorfmeister and Neher \cite{DN} then employed isoparametric triple systems \cite{DN1}, which are algebraic in nature, to prove that Condition A alone implies the isoparametric hypersurface is of OT-FKM type. Their proof for the case of multiplicity pairs $\{3,4\}$ and $\{7,8\}$ rests on a fairly involved algebraic classification result \cite{Mc} about composition triples. In light of the classification \cite{CCJ} that leaves only the four exceptional multiplicity pairs $\{4,5\},\{3,4\},\{7,8\}$ and $\{6,9\}$ unsettled, it appears that Condition A may hold the key to the classification when the multiplicity pairs are $\{3,4\}$ and $\{7,8\}$. Thus Condition A deserves to be scrutinized and understood more thoroughly from different angles. In this paper, we give a fairly short and rather straightforward proof of the result of Dorfmeister and Neher, with emphasis on the multiplicity pairs $\{3,4\}$ and $\{7,8\}$, based on more geometric considerations. We make it explicit and apparent that the octonian algebra governs the underlying isoparametric structure.