Normal Scalar Curvature Inequality on the Focal Submanifolds of Isoparametric Hypersurfaces

Abstract

An isoparametric hypersurface in unit spheres has two focal submanifolds. Condition A plays a crucial role in the classification theory of isoparametric hypersurfaces in [CCJ07], [Chi16] and [Miy13]. This paper determines CA, the set of points with Condition A in focal submanifolds. It turns out that the points in CA reach an upper bound of the normal scalar curvature ρ ⊥ (sharper than that in DDVV inequality [GT08], [Lu11]). We also determine the sets CP (points with parallel second fundamental form) and CE (points with Einstein condition), which achieve two lower bounds of ρ ⊥ .

Some generalizations of the DDVV and BW inequalities

“…Submanifolds achieving the equality everywhere are called Wintgen ideal submanifolds which are not classified so far (cf. [6,7,31,17]). Besides of this original geometric background, the DDVV inequality has also many important applications, for example, in deriving Simons-type inequality and pinching results for the second fundamental form in submanifold geometry (cf.…”

“…(1) What is the expectation of the commutators of random matrices in certain categories like GOE, GUE, and GSE? (2) Find more DDVV-type inequalities for matrices, Lie algebras or operators lying in certain subspaces of special interest like spaces of austere matrices (see for a special example in [17]). (3) Whether the nonnegative polynomial defined by the DDVV-type inequalities (by…”

Isoparametric polynomials and sums of squares

“…For example, they are austere submanifolds (thus minimal) with constant principal curvatures independent of the choice of normal directions (cf. [25,23]). For the cases g = 4 (resp.…”

Extrinsic geometry of the Gromoll-Meyer sphere

“…The isoparametric theory initiated from the study of E. Cartan in real space forms in 1940s, which caught much caution recently as the accomplishment of the classifications of isoparametric hypersurfaces in the unit sphere ( [CCJ07], [Chi13], [Miy13], [Miy16], [Chi16]). The applications of isoparametric theory are more and more abundant, see for example [TY13], [QTY13], [TXY14] and [GTY18].…”