Principal Curvatures of Isoparametric Hypersurfaces in Cp
نویسنده
چکیده
Let M be an isoparametric hypersurface in CPn, and M the inverse image of M under the Hopf map. By using the relationship between the eigenvalues of the shape operators of M and M , we prove that M is homogeneous if and only if either g or l is constant, where g is the number of distinct principal curvatures of M and l is the number of non-horizontal eigenspaces of the shape operator on M .
منابع مشابه
PARA-BLASCHKE ISOPARAMETRIC HYPERSURFACES IN THE UNIT SPHERE Sn+1(1) (II)
Let D = A + λB be the para-Blaschke tensor of the immersion x, where λ is a constant, A and B are the Blaschke tensor and the Möbius second fundamental form of x. A hypersurface x : M 7→ S(1) in the unit sphere S(1) without umbilical points is called a para-Blaschke isoparametric hypersurface if the Möbius form Φ vanishes identically and all of its para-Blaschke eigenvalues are constants. In [1...
متن کامل$L_1$-Biharmonic Hypersurfaces in Euclidean Spaces with Three Distinct Principal Curvatures
Chen's biharmonic conjecture is well-known and stays open: The only biharmonic submanifolds of Euclidean spaces are the minimal ones. In this paper, we consider an advanced version of the conjecture, replacing $Delta$ by its extension, $L_1$-operator ($L_1$-conjecture). The $L_1$-conjecture states that any $L_1$-biharmonic Euclidean hypersurface is 1-minimal. We prove that the $L_1$-conje...
متن کامل$L_k$-biharmonic spacelike hypersurfaces in Minkowski $4$-space $mathbb{E}_1^4$
Biharmonic surfaces in Euclidean space $mathbb{E}^3$ are firstly studied from a differential geometric point of view by Bang-Yen Chen, who showed that the only biharmonic surfaces are minimal ones. A surface $x : M^2rightarrowmathbb{E}^{3}$ is called biharmonic if $Delta^2x=0$, where $Delta$ is the Laplace operator of $M^2$. We study the $L_k$-biharmonic spacelike hypersurfaces in the $4$-dimen...
متن کاملA Note on the Paper ”isoparametric Hypersurfaces with Four Principal Curvatures”
In [6], employing commutative algebra, we showed that if the number of principal curvatures is 4 and if the multiplicities m1 and m2 of the principal curvatures satisfy m2 ≥ 2m1 − 1, then the isoparametric hypersurface is of the type constructed by Ozeki-Takeuchi and Ferus-Karcher-Münzner [18], [11]. This leaves only four multiplicity pairs (m1, m2) = (3, 4), (4, 5), (6, 9) and (7, 8) unsettled...
متن کاملTriality and the Normal Sections of Cartan’s Isoparametric Hypersurfaces
The present paper is devoted to study the algebraic sets of normal sections of the so called Cartan’s isoparametric hypersurfaces MR, MC, MH and MO of complete flags in the projective planes RP , CP , HP 2 and OP . It presents a connection between “normed trialities” and the polynomial defining the algebraic sets of normal sections of these hypersurfaces. It contains also a geometric and topolo...
متن کامل