Helical immersions into a unit sphere

Sakamoto, Kunio

doi:10.1007/bf01456411

Cited by 37 publications

(22 citation statements)

References 11 publications

(12 reference statements)

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“…A Euclidean submanifold M is called helical if geodesics of M , considered as curves in E m , have all Frenet curvatures constant and independent of the chosen geodesic. Helical immersions have been studied extensively (see, for example, [18,24]). …”

Section: Lemma 34 An Isotropic Submanifold M Is Constant Isotropic mentioning

confidence: 99%

The Contact Number of a Euclidean Submanifold

Chen¹,

Li²

2004

Proceedings of the Edinburgh Mathematical Society

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We introduce an invariant, called the contact number, associated with each Euclidean submanifold. We show that this invariant is, surprisingly, closely related to the notions of isotropic submanifolds and holomorphic curves. We are able to establish a simple criterion for a submanifold to have any given contact number. Moreover, we completely classify codimension-2 submanifolds with contact number 3. We also study surfaces in E 6 with contact number 4. As an immediate consequence, we obtain the first explicit examples of non-spherical pseudo-umbilical surfaces in Euclidean spaces.

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Section: Lemma 34 An Isotropic Submanifold M Is Constant Isotropic mentioning

confidence: 99%

The Contact Number of a Euclidean Submanifold

Chen¹,

Li²

2004

Proceedings of the Edinburgh Mathematical Society

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“…timelike) geodesic γ of M into a helix of type Λ which is independent of γ. This notion is a generalization in pseudoRiemannian geometry of that in Sakamoto [14]. In [9], the second author proves the following conditions are equivalent in the case that the domain of f is indefinite.…”

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confidence: 91%

Extremal Lorentzian surfaces with null $r$-planar geodesics in space forms

Hasegawa¹,

Miura²

2015

Tohoku Math. J. (2)

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We show a congruence theorem for oriented Lorentzian surfaces with horizontal reflector lifts in pseudo-Riemannian space forms of neutral signature. As a corollary, a characterization theorem can be obtained for the Lorentzian Boruvka spheres, that is, a full real analytic null r-planar geodesic immersion with vanishing mean curvature vector field is locally congruent to the Lorentzian Boruvka sphere in a 2r-dimensional space form of neutral signature.

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“…In case (B), we see that the second fundamental form is parallel and / is 2-planar geodesic. In [19], we classified all 2-planar geodesic immersions into 5m(l). Next, let us consider cases (C) and (D) where the immersions are helical of orders 3 and 4, respectively.…”

Section: Frenet Curvaturesmentioning

confidence: 99%

Constant isotropic submanifolds with 4-planar geodesics

Pak¹,

Sakamoto²

1988

Trans. Amer. Math. Soc.

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ABSTRACT. Let / be an isometric immersion of a Riemannian manifold M into M. We prove that if / is constant isotropic, 4-planar geodesic and M is a Euclidean sphere, then M is isometric to one of compact symmetric spaces of rank equal to one and / is congruent to a direct sum of standard minimal immersions. We also determine constant isotropic, 4-planar geodesic, totally real immersions into a complex projective space of constant holomorphic sectional curvature. Introduction.In [15], O'Neill studied isotropic immersions and showed that if the difference between the sectional curvatures of the submanifold and the ambient manifold is constant for any plane section tangent to the submanifold, then the codimension must be high compared with the dimension of the submanifold unless the immersion is totally geodesic or umbilical. The class of (constant) isotropic immersions (for instance, into a sphere) seems to be too large to classify them. For example, the composition of two isotropic immersions is also isotropic and for two given isotropic immersions fy and f2 into 5P(1) and 5a(l), respectively, their "direct sum" (cos8fy,sin8f2) into 5p+,+1(l) is also isotropic for any 9. Furthermore, it is well known that there are many constant isotropic immersions in equivariant isometric immersions of homogeneous spaces.An isometric immersion /: A_ -► M is said to be 4-planar geodesic if, for each geodesic r of M, the curve / o 7 is locally contained in a 4-dimensional totally geodesic submanifold of M. The class of 4-planar geodesic immersions (into a sphere) is also large. At least, an isometric immersion of a surface into a 4-dimensional Riemannian manifold is always 4-planar geodesic.In this paper, we try to classify immersions in the intersection of these classes. In particular, we are able to classify constant isotropic, 4-planar geodesic (resp. and totally real) immersions of connected, complete Riemannian manifolds into a unit sphere (resp. complex projective space of constant holomorphic sectional curvature 4).In §1, we give basic equations used later and some definitions. In §2, we prove that if / is a constant isotropic, 4-planar geodesic (resp. and totally real) immersion into a real space form (resp. CPq(c)), then / is a helical immersion of order < 4. Here we recall that an isometric immersion /: M -» M is called a helical immersion

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Helical immersions into a unit sphere

Cited by 37 publications

References 11 publications

The Contact Number of a Euclidean Submanifold

The Contact Number of a Euclidean Submanifold

Extremal Lorentzian surfaces with null $r$-planar geodesics in space forms

Constant isotropic submanifolds with 4-planar geodesics

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