Bi-order real hypersurfaces in a complex projective space

Udagawa, Seiichi

doi:10.2996/kmj/1138037413

Cited by 5 publications

(19 citation statements)

References 16 publications

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“…We now examine which of the Hopf hypersurfaces with constant principal curvatures are of 2-type. This has been already considered by Udagawa for hypersurfaces of CP m [36]. Although our argument is different from Udagawa's and relies on the analysis of the conditions (E 1 ) -(E 4 ), rather than on the matrix representation of the immersion in H (1) (m + 1), it partly overlaps Udagawa's investigation and reaches the same classification for 2-type CMC real hypersurfaces in CP m of class A.…”

Section: The Classification Of 2-type Hopf Hypersurfaces Of Cq M (4c)supporting

confidence: 51%

“…Using the complete list of such hypersurfaces available in [32], [24], [20], [3], [4], one obtains a classification of constant-mean-curvature (CMC) hypersurfaces whose Chen-type is 2. This has been already attempted by Udagawa [36] for hypersurfaces of CP m (4), and for hypersurfaces of CH m (−4) see below. Udagawa's classification in CP m , however, is incomplete (see below).…”

Section: Hopf Hypersurfaces Of 2-type Have Constant Principal Curvaturesmentioning

confidence: 98%

“…Let L be the vector field in H (1) (m + 1) along M n , represented by the left hand side of the equation (36) and let X be an arbitrary tangential vector field of M n . Then 10…”

Section: Hopf Hypersurfaces Of 2-type Have Constant Principal Curvaturesmentioning

confidence: 99%

“…Conversely, if (44) holds thenx satisfies the polynomial equation (36) in the Laplacian of the form P (∆)(x −x 0 ) = 0, with P (t) = t 2 −pt+q. According to a result of Chen and Petrovic [12] if such polynomial has simple real roots the submanifold is of 2-type (if not already of 1-type).…”

Section: Hopf Hypersurfaces Of 2-type Have Constant Principal Curvaturesmentioning

confidence: 99%

“…class-A 1 hypersurfaces) in CP m which are mass-symmetric and of 2-type, whereas we prove that a geodesic hypersphere of radius cot −1 (1/ √ m) exactly has these properties. Because of these erroneous claims, all three theorems of [36] are deficient in one way or another.…”

Section: Introductionmentioning

confidence: 99%

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Hopf hypersurfaces of low type in non-flat complex space forms

Dimitrić¹

2011

Kodai Math. J.

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We classify Hopf hypersurfaces of non-flat complex space forms CP m (4) and CH m (−4), denoted jointly by CQ m (4c), that are of 2-type in the sense of B. Y. Chen, via the embedding into a suitable (pseudo) Euclidean space of Hermitian matrices by projection operators. This complements and extends earlier classifications by Martinez-Ros (minimal case) and Udagawa (CMC case), who studied only hypersurfaces of CP m and assumed them to have constant mean curvature instead of being Hopf. Moreover, we rectify some claims in Udagawa's paper to give a complete classification of constant-mean-curvature-hypersurfaces of 2-type. We also derive a certain characterization of CMC Hopf hypersurfaces which are of 3-type and masssymmetric in a naturally-defined hyperquadric containing the image of CQ m (4c) via these embeddings. The classification of such hypersurfaces is done in CQ 2 (4c), under an additional assumption in the hyperbolic case that the mean curvature is not equal to ±2/3. In the process we show that every standard example of class B in CQ m (4c) is mass-symmetric and we determine its Chen-type.

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Section: The Classification Of 2-type Hopf Hypersurfaces Of Cq M (4c)supporting

confidence: 51%

Section: Hopf Hypersurfaces Of 2-type Have Constant Principal Curvaturesmentioning

confidence: 98%

“…Let L be the vector field in H (1) (m + 1) along M n , represented by the left hand side of the equation (36) and let X be an arbitrary tangential vector field of M n . Then 10…”

Section: Hopf Hypersurfaces Of 2-type Have Constant Principal Curvaturesmentioning

confidence: 99%

Section: Hopf Hypersurfaces Of 2-type Have Constant Principal Curvaturesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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