No abstract uploaded!

The classification work [5], [8] 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 S19 is homogeneous, and, moreover, an isoparametric hypersurface with four principal curvatures and multiplicities {6, 9} in S31 is either the inhomogeneous one constructed by Ferus, Karcher, and M¨unzner, 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¨unzner and the one with multiplicities {4, 5} stands alone. The quaternion and the octonion algebras play a fundamental role in their geometric structures. A unifying theme in [5], [8], 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], [8] and extend in this sequel is instrumental. The classification leaves only the case of multiplicity pair {7, 8} open.

H. Hopf showed that the only constant mean curvature sphere S2 immersed in R3 is the round sphere. The K¨ahler framework is an adequate approach to generalize Hopf’ s theorem to higher dimensions. When ϕ : M → Rn is an isometric immersion from a K¨ahler manifold, the complexified second fundamental form α splits according to types. The (1, 1) part of the second fundamental form plays the role of the mean curvature for surfaces and will be called the pluri-mean curvature pmc. Therefore isometric immersions with parallel pluri-mean curvature (ppmc isometric immersions) generalize in a natural way the cmc immersions. It is a standard fact that R8 is the smallest space where CP2 can be embedded. The aim of this work is to generalize Hopf’s theorem proving in particular that the only ppmc isometric immersion from CP2 into R8 is the standard immersion.

No abstract uploaded!

Suppose A and B are normed division algebras, i.e. R,C,H or O, we introduce and study Grassmannians of linear subspaces in (A ⊗ B)n which are complex/Lagrangian/maximal isotropic with respect to natural two tensors on (A ⊗ B)n. We show that every irreducible compact symmetric space must be one of these Grassmannian spaces, possibly up to a finite cover. This gives a simple and uniform description of all compact symmetric spaces. This generalizes the Tits magic square description for simple Lie algebras to compact symmetric spaces.