Isoparametric hypersurfaces with four principal curvatures, II

AbstractIn this sequel to an earlier article, employing more commutative algebra than previously, we show that an isoparametric hypersurface with four principal curvatures and multiplicities (3,4) inS15is one constructed by Ozeki and Takeuchi and Ferus, Karcher, and Münzner, referred to collectively asof OT-FKM type. In fact, this new approach also gives a considerably simpler proof, both structurally and technically, that an isoparametric hypersurface with four principal curvatures in spheres with the multiplicity constraintm2≥2m1-1 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 in that it cannot be of OT-FKM type. A by-product of this new approach is that we see that Condition B, introduced by Ozeki and Takeuchi 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.

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Isoparametric hypersurfaces with four principal curvatures, II

Nagoya Mathematical Journal ◽

10.1215/00277630-1431813 ◽

2011 ◽

Vol 204 ◽

pp. 1-18 ◽

Cited By ~ 17

Author(s):

Quo-Shin Chi

Keyword(s):

Commutative Algebra ◽

Quaternion Algebra ◽

Type A ◽

Isoparametric Hypersurface ◽

Principal Curvatures ◽

Octonion Algebra ◽

Isoparametric Hypersurfaces ◽

New Approach ◽

Condition B

AbstractIn this sequel to an earlier article, employing more commutative algebra than previously, we show that an isoparametric hypersurface with four principal curvatures and multiplicities (3,4) in S15 is one constructed by Ozeki and Takeuchi and Ferus, Karcher, and Münzner, referred to collectively as of OT-FKM type. In fact, this new approach also gives a considerably simpler proof, both structurally and technically, that an isoparametric hypersurface with four principal curvatures in spheres with the multiplicity constraint m2≥ 2m1 -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 in that it cannot be of OT-FKM type. A by-product of this new approach is that we see that Condition B, introduced by Ozeki and Takeuchi 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.

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PARA-BLASCHKE ISOPARAMETRIC HYPERSURFACES IN A UNIT SPHERE Sn + 1(1)

Glasgow Mathematical Journal ◽

10.1017/s001708951200016x ◽

2012 ◽

Vol 54 (3) ◽

pp. 579-597 ◽

Cited By ~ 3

Author(s):

SHICHANG SHU ◽

BIANPING SU

Keyword(s):

Fundamental Form ◽

Unit Sphere ◽

Second Fundamental Form ◽

Isoparametric Hypersurface ◽

Principal Curvatures ◽

Isoparametric Hypersurfaces ◽

Blaschke Tensor ◽

Blaschke Isoparametric Hypersurface ◽

Full Classification ◽

Möbius Second Fundamental Form

AbstractLet A = ρ2∑i,jAijθi ⊗ θj and B = ρ2∑i,jBij θi ⊗ θj be the Blaschke tensor and the Möbius second fundamental form of the immersion x. Let D = A + λB be the para-Blaschke tensor of x, where λ is a constant. If x: Mn ↦ Sn + 1(1) is an n-dimensional para-Blaschke isoparametric hypersurface in a unit sphere Sn + 1(1) and x has three distinct Blaschke eigenvalues one of which is simple or has three distinct Möbius principal curvatures one of which is simple, we obtain the full classification theorems of the hypersurface.

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Isoparametric Hypersurfaces with four Principal Curvatures Revisited

Nagoya Mathematical Journal ◽

10.1017/s0027763000026064 ◽

2009 ◽

Vol 193 ◽

pp. 129-154 ◽

Cited By ~ 9

Author(s):

Quo-Shin Chi

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Fundamental Form ◽

Isoparametric Hypersurface ◽

Geometric Meaning ◽

Principal Curvatures ◽

Expansion Formula ◽

Isoparametric Hypersurfaces ◽

Focal Submanifold

AbstractThe classification of isoparametric hypersurfaces with four principal curvatures in spheres in [2] hinges on a crucial characterization, in terms of four sets of equations of the 2nd fundamental form tensors of a focal submanifold, of an isoparametric hypersurface of the type constructed by Ferus, Karcher and Münzner. The proof of the characterization in [2] is an extremely long calculation by exterior derivatives with remarkable cancellations, which is motivated by the idea that an isoparametric hypersurface is defined by an over-determined system of partial differential equations. Therefore, exterior differentiating sufficiently many times should gather us enough information for the conclusion. In spite of its elementary nature, the magnitude of the calculation and the surprisingly pleasant cancellations make it desirable to understand the underlying geometric principles.In this paper, we give a conceptual, and considerably shorter, proof of the characterization based on Ozeki and Takeuchi’s expansion formula for the Cartan-Münzner polynomial. Along the way the geometric meaning of these four sets of equations also becomes clear.

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Conformal isoparametric hypersurfaces with two distinct conformal principal curvatures in conformal space

Science China Mathematics ◽

10.1007/s11425-009-0206-4 ◽

2010 ◽

Vol 53 (4) ◽

pp. 953-965 ◽

Cited By ~ 6

Author(s):

ChangXiong Nie ◽

TongZhu Li ◽

YiJun He ◽

ChuanXi Wu

Keyword(s):

Conformal Space ◽

Principal Curvatures ◽

Isoparametric Hypersurfaces

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Isoparametric hypersurfaces with less than four principal curvatures in a sphere

Colloquium Mathematicum ◽

10.4064/cm105-1-13 ◽

2006 ◽

Vol 105 (1) ◽

pp. 143-148 ◽

Cited By ~ 2

Author(s):

Toshiaki Adachi ◽

Sadahiro Maeda

Keyword(s):

Principal Curvatures ◽

Isoparametric Hypersurfaces

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A Class of Special Hypersurfaces in Real Space Forms

Journal of Function Spaces ◽

10.1155/2016/8796938 ◽

2016 ◽

Vol 2016 ◽

pp. 1-6

Author(s):

Yan Zhao ◽

Ximin Liu

Keyword(s):

Real Space ◽

Principal Curvatures ◽

Space Forms ◽

Isoparametric Hypersurfaces ◽

Real Space Forms ◽

Constant Principal Curvatures

We define the generalized golden- and product-shaped hypersurfaces in real space forms. A hypersurfaceMin real space formsRn+1,Sn+1, andHn+1is isoparametric if it has constant principal curvatures. Based on the classification of isoparametric hypersurfaces, we obtain the whole families of the generalized golden- and product-shaped hypersurfaces in real space forms.

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Divergence-preserving geodesic transformations

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500027347 ◽

1998 ◽

Vol 128 (6) ◽

pp. 1309-1323

Author(s):

E. García-Río ◽

L. Vanhecke

Keyword(s):

Principal Curvatures ◽

Isoparametric Hypersurfaces ◽

Hopf Hypersurfaces ◽

Geodesic Spheres ◽

Constant Principal Curvatures ◽

Volume Preserving ◽

Special Classes

We discuss divergence- and volume-preserving geodesic transformations with respect to submanifolds and in particular, with respect to hypersurfaces. We use these transformations to derive characterisations of special classes of hypersurfaces such as isoparametric hypersurfaces and Hopf hypersurfaces with constant principal curvatures. Furthermore, we consider divergence-preserving geodesic transformations with respect to geodesic spheres.

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On the topology of isoparametric hypersurfaces with four distinct principal curvatures

Proceedings of the American Mathematical Society ◽

10.1090/s0002-9939-99-04490-1 ◽

1999 ◽

Vol 127 (1) ◽

pp. 259-264 ◽

Cited By ~ 6

Author(s):

Fuquan Fang

Keyword(s):

Principal Curvatures ◽

Isoparametric Hypersurfaces

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A new approach on electromagnetism with dual number coefficient octonion algebra

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887816300130 ◽

2016 ◽

Vol 13 (09) ◽

pp. 1630013 ◽

Cited By ~ 8

Author(s):

Bhupesh Chandra Chanyal ◽

Sunil Kumar Chanyal ◽

Özcan Bektaş ◽

Salim Yüce

Keyword(s):

Maxwell Equations ◽

Wave Equations ◽

Space Time ◽

Internal Space ◽

Field Equations ◽

Octonion Algebra ◽

New Approach ◽

Dual Number ◽

Symmetrical Form ◽

Unit Formula

Dual number coefficient octonion (DNCO) is one of the kind of octonion, it has 16 components with an additional dual unit [Formula: see text]. Starting with DNCO algebra, we develop the generalized electromagnetic field equations of dyons regarding the DNCOS spaces, which has two octonionic space-times namely the octonionic internal space-time and the octonionic external space-time. Besides, the generalized four-potential components of dyons have been expressed with respect to the dual octonion form. Furthermore, we obtain the symmetrical form of Dirac–Maxwell equations, and the generalized potential wave equations for dyons in terms of the dual octonion. Finally, we conclude that dual octonion formulation is compact and simpler like octonion formulation.

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