The Contact Number of a Euclidean Submanifold

Chen, Bang‐Yen; Li, Shijie

doi:10.1017/s0013091503000038

Cited by 12 publications

(23 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Remark 5.5 The last result is not true for n = 2. In fact, every non-planar holomorphic curve with respect to some orthogonal complex structure on R 4 is a minimal isotropic and non-constant isotropic surface (see [12]). , q), t) for any p, q ∈ N n and t ∈ R + 0 .…”

Section: (B) If N ≥ 3 φ Is Pseudo-umbilical If and Only If N N S Is mentioning

confidence: 99%

See 1 more Smart Citation

Isotropic submanifolds of pseudo-Riemannian spaces

Cabrerizo

Fernández

Gómez

2012

Journal of Geometry and Physics

View full text Add to dashboard Cite

The family of all the submanifolds of a given Riemannian or pseudo-Riemannian manifold is large enough to classify them into some interesting subfamilies such as minimal (maximal), totally geodesic, Einstein, etc. Most of these have been extensively studied by many authors, but as far as we know, no paper has hitherto been published on the class of isotropic submanifolds. The purpose of this paper is therefore to gain a better understanding of this interesting class of submanifolds that arise naturally in mathematics and physics by studying their relationships with other closely distinguised families.Mathematics Subject Classification: 53C25, 53C40

show abstract

Section: (B) If N ≥ 3 φ Is Pseudo-umbilical If and Only If N N S Is mentioning

confidence: 99%

“…defines a constant isotropic immersion for some suitable constant c k [12]. It is known [6] that ϕ k (N n ) is minimal in a certain Euclidean sphere of R m .…”

Section: Corollary 53 Let φ : N N S → M M ν (C) Be An Isotropic and mentioning

confidence: 99%

Isotropic submanifolds of pseudo-Riemannian spaces

Cabrerizo

Fernández

Gómez

2012

Journal of Geometry and Physics

View full text Add to dashboard Cite

show abstract

“…As for constant isotropic submanifolds in the Euclidean space [4], by using (3) in Lemma 3.5 we have the following characterization of the constant pseudo-isotropic immersions in the pseudo-Euclidean space. …”

Section: Remark 34mentioning

confidence: 99%

Rigidity of pseudo-isotropic immersions

Cabrerizo

Fernández

Gómez

2009

Journal of Geometry and Physics

View full text Add to dashboard Cite

Several notions of isotropy of a (pseudo)Riemannian manifold have been introduced in the literature, in particular, the concept of pseudo-isotropic immersion. The aim of this paper is to look more closely at this notion of pseudoisotropy and then to study the rigidity of this class of immersion into the pseudoEuclidean space. It is worth pointing out that we first obtain a characterization of the pseudo-isotropy condition by using tangent vectors of any causal character. Then, rigidity theorems for pseudo-isotropic immersions are proved, and in particular, some well known results for the Riemannian case arise. Later, we bring together the notions of pseudo-isotropy, intrinsically and extrinsically isotropic manifolds, and prove interesting relations among them. Finally, we pay special attention to the case of codimension two Lorentz surfaces.

show abstract

“…Moreover, formulas (8) and (9) in Lemma 3.3 can be also obtained. As for constant isotropic submanifolds in the Euclidean space [5], we have the following characterization of the constant pseudo-isotropic submanifolds in the semi-Euclidean space. …”

Section: Lemma 33 a Pseudo-riemannian Submanifoldmentioning

confidence: 99%

“…Chen and S.-J. Li introduced and studied the notion of contact number c (M ) of a Euclidean submanifold in [5], and they proved that the contact number is closely related with the notions of isotropic submanifolds and holomorphic curves. In particular, a surface in the Euclidean space R 4 has contact number 3 if an only it is a non-planar holomorphic curve with respect to some orthogonal complex structure on R 4 .…”

Section: Introductionmentioning

confidence: 99%

The Contact Number of a Pseudo-Euclidean Submanifold

Cabrerizo¹,

́andez²,

́omez³

2008

Taiwanese J. Math.

View full text Add to dashboard Cite

Abstract. In this paper we define the contact number of a pseudo-Riemannian submanifold into the pseudo-Euclidean space, and prove that this contact number is closely related to the notion of pseudo-isotropic submanifold. We give a classification of hypersurfaces into the pseudo-Euclidean space with contact number at least 3. A classification of the complete spacelike codimension-2 submanifolds of the Lorentz-Minkowski space with contact number at least 3 is also obtained.

show abstract

The Contact Number of a Euclidean Submanifold

Cited by 12 publications

References 18 publications

Isotropic submanifolds of pseudo-Riemannian spaces

Isotropic submanifolds of pseudo-Riemannian spaces

Rigidity of pseudo-isotropic immersions

The Contact Number of a Pseudo-Euclidean Submanifold

Contact Info

Product

Resources

About