A new look at Condition A

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

Isoparametric hypersurfaces with four principal curvatures, II

In this sequel, employing more commutative algebra than that explored in \cite{CCJ}, we show that an isoparametric hypersurface with four principal curvatures and multiplicities $(3,4)$ in $S^{15}$ is one constructed by Ozeki-Takeuchi \cite[I]{OT} and Ferus-Karcher-M\"unzner \cite{FKM}, referred to collectively as of OT-FKM type. In fact, this new approach also gives a considerably simpler, both structurally and technically, proof \cite{CCJ} that an isoparametric hypersurface with four principal curvatures in spheres with the multiplicity constraint $m_2\geq 2m_1-1$ is of OT-FKM type, which left unsettled exactly the four anomalous multiplicity pairs $(4,5),(3,4),(7,8)$ and $(6,9)$, where the last three are closely tied, respectively, with the quaternion algebra, the octonion algebra and the complexified octonion algebra, whereas the first stands alone by itself in that it cannot be of OT-FKM type. A byproduct of this new approach is that we see that Condition B, introduced by Ozeki and Takeuchi \cite[I]{OT} in their construction of inhomogeneous isoparametric hypersurfaces, naturally arises. The cases for the multiplicity pairs $(4,5),(6,9)$ and $(7,8)$ remain open now.Comment: This is a slightly revised version of arXiv: 1002.1345. Correction of typos is made in Section 2 and the first displayed formula following Formula (20

Ideal theory and classification of isoparametric hypersurfaces

The classification of isoparametric hypersurfaces with four principal curvatures in the sphere interplays in a deep fashion with commutative algebra, whose abstract and comprehensive nature might obscure a differential geometer's insight into the classification problem that encompasses a wide spectrum of geometry and topology. In this paper, we make an effort to bridge the gap by walking through the important part of commutative algebra central to the classification of such hypersurfaces, such that all the essential ideal-theoretic ingredients are laid out in a way as much intuitive, motivating and geometric with rigor maintained as possible. We then explain how we developed the technical side of the entailed ideal theory, pertinent to isoparametric hypersurfaces with four principal curvatures, for the classification done in our papers~\cite{CCJ},~\cite{Ch1} and~\cite{Ch3}.Comment: 27 pages. The article has been submitted to the Proceedings of the 2013 Midwest Geometry Conferenc

Isoparametric hypersurfaces with four principal curvatures, III

The classification work [5], [9] left unsettled only those anomalous isoparametric hypersurfaces with four principal curvatures and multiplicity pair $\{4,5\},\{6,9\}$ or $\{7,8\}$ in the sphere. By systematically exploring the ideal theory in commutative algebra in conjunction with the geometry of isoparametric hypersurfaces, we show that an isoparametric hypersurface with four principal curvatures and multiplicities $\{4,5\}$ in $S^{19}$ is homogeneous, and, moreover, an isoparametric hypersurface with four principal curvatures and multiplicities $\{6,9\}$ in $S^{31}$ is either the inhomogeneous one constructed by Ferus, Karcher and M\"{u}nzner, or the one that is homogeneous. This classification reveals the striking resemblance between these two rather different types of isoparametric hypersurfaces in the homogeneous category, even though the one with multiplicities $\{6,9\}$ is of the type constructed by Ferus, Karcher and M\"{u}nzner and the one with multiplicities $\{4,5\}$ stands alone by itself. The quaternion and the octonion algebras play a fundamental role in their geometric structures. A unifying theme in [5]. [9] and the present sequel to them is Serre's criterion of normal varieties. Its technical side pertinent to our situation that we developed in [5], [9] and extend in this sequel is instrumental. The classification leaves only the case of multiplicity pair $\{7,8\}$ open.Comment: 35 pages. In the abstract and several other places, Ozeki and Takeuchi are replaced by Ferus, Karcher and M\"{u}nzner, who constructed the inhomogeneous example of multiplicities (6,9). The paper is to replace an earlier version with the same title, in which the case (4,5) was handled. In this version the case (6,9) is also settled. Only the case of multiplicities (7,8) remains open no

Orthogonal multiplications of type [3,4,p],p≤12[3,4,p], p\leq 12

We describe the moduli space of orthogonal multiplications of type $[3,4,p], p\leq 12,$ and its application to the hypersurface theory.Comment: 27 page

Exotic Holonomy on Moduli Spaces of Rational Curves

Bryant \cite{Br} proved the existence of torsion free connections with exotic holonomy, i.e. with holonomy that does not occur on the classical list of Berger \cite{Ber}. These connections occur on moduli spaces \Y of rational contact curves in a contact threefold \W. Therefore, they are naturally contained in the moduli space $\Z$ of all rational curves in \W. We construct a connection on $\Z$ whose restriction to \Y is torsion free. However, the connection on $\Z$ has torsion unless both \Y and $\Z$ are flat. We also show the existence of a new exotic holonomy which is a certain sixdimensional representation of \Sl \times \Sl. We show that every regular $H_3$-connection (cf. \cite{Br}) is the restriction of a unique connection with this holonomy.Comment: 30 pages, AMS-TeX, uses picte

Structure of minimal 2-spheres of constant curvature in the complex hyperquadric

In this paper, the singular-value decomposition theory of complex matrices is explored to study constantly curved 2-spheres minimal in both $\mathbb{C}P^n$ and the hyperquadric of $\mathbb{C}P^n$. The moduli space of all those noncongruent ones is introduced, which can be described by certain complex symmetric matrices modulo an appropriate group action. Using this description, many examples, such as constantly curved holomorphic 2-spheres of higher degree, nonhomogenous minimal 2-spheres of constant curvature, etc., are constructed. Uniqueness is proven for the totally real constantly curved 2-sphere minimal in both the hyperquadric and $\mathbb{C}P^n$.Comment: 30 pages, 2 figure

Dupin hypersurfaces with four principal curvatures, II

If $M$ is an isoparametric hypersurface in a sphere $S^n$ with four distrinct principal curvatures, then the principal curvatures $\kappa_1,...,\kappa_4$ can be ordered so that their multiplicities satisfy $m_1=m_2$ and $m_3=m_4$, and the cross-ratio $r$ of the principal curvatures (the Lie curvature) equals -1. In this paper, we prove that if $M$ is an irreducible connected proper Dupin hypersurface in $\R^n$ (or $S^n$) with four distinct principal curvatures with multiplicities $m_1=m_2 \geq 1$ and $m_3=m_4=1$, and constant Lie curvature $r=-1$, then $M$ is equivalent by Lie sphere transformation to an isoparametric hypersurface in a sphere. This result remains true if the assumption of irreducibility is replaced by compactness and $r$ is merely assumed to be constant

On Kuiper's conjecture

We prove that any connected proper Dupin hypersurface in $\R^n$ is analytic algebraic and is an open subset of a connected component of an irreducible algebraic set. We prove the same result for any connected non-proper Dupin hypersurface in $\R^n$ that satisfies a certain finiteness condition. Hence any taut submanifold M in $\R^n$, whose tube $M_\epsilon$ satisfies this finiteness condition, is analytic algebraic and is a connected component of an irreducible algebraic set. In particular, we prove that every taut submanifold of dimension $m \leq 4$ is algebraic.Comment: 43 page

Isoparametric hypersurfaces with four principal curvatures, IV

We prove that an isoparametric hypersurface with four principal curvatures and multiplicity pair $(7,8)$ is either the one constructed by Ozeki and Takeuchi, or one of the two constructed by Ferus, Karcher, and M\"{u}nzner. This completes the classification of isoparametric hypersurfaces in spheres that \'{E}. Cartan initiated in the late 1930s.Comment: 68 pages. Appendix II, which handles the anomalous case in an ad hoc fashion, in the previous version is now removed with the availability of the preprint entitled "Orthogonal multiplications of type $[3, 4, p], p\leq 12$", arXiv:1705.04762. Accordingly, the Introduction and Section 7, up to and including Remark 7.3., are slightly reworded with no change in all conclusion