Infinitesimal variations  of submanifolds

Dajczer, Marcos; Jimenez, M. I.

doi:10.21711/217504322021/em35

Cited by 2 publications

(4 citation statements)

References 23 publications

(39 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Therefore, the fact that Ā and B satisfy (22) implies that B * and A h also satisfy (22) with respect to ḡ. Since, in addition, tr B * = 0, it follows from Proposition 8 of [11], together with Theorem 4.7 in [4] (also stated in [11] as Theorem 2), that h is locally infinitesimally Bonnet bendable, hence isothermic by Proposition 9 of [11].…”

Section: Proof Of Theoremmentioning

confidence: 90%

“…The study of hypersurfaces f : M n → R n+1 , n ≥ 3, that admit nontrivial variations preserving the induced metric goes back to Sbrana [15] and Cartan [1] (see also [8] or Chapter 11 of [9]), whereas the hypersurfaces that admit non-trivial infinitesimal variations were investigated by Sbrana [14] (see also [6], Chapter 14 of [9] and [4]). We point out that the latter class turns out to be much larger than the former.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Infinitesimally Bonnet Bendable Hypersurfaces

Jimenez

Tojeiro

2023

J Geom Anal

View full text Add to dashboard Cite

Li, Ma and Wang have provided in [13] a partial classification of the so-called Moebius deformable hypersurfaces, that is, the umbilicfree Euclidean hypersurfaces f : M n → R n+1 that admit non-trivial deformations preserving the Moebius metric. For n ≥ 5, the classification was completed by the authors in [12]. In this article we obtain an infinitesimal version of that classification. Namely, we introduce the notion of an infinitesimal Moebius variation of an umbilic-free immersion f : M n → R m into Euclidean space as a one-parameter family of immersions f t : M n → R m , with t ∈ (−ǫ, ǫ) and f 0 = f , such that the Moebius metrics determined by f t coincide up to the first order. Then we characterize isometric immersions f : M n → R m of arbitrary codimension that admit a non-trivial infinitesimal Moebius variation among those that admit a non-trivial conformal infinitesimal variation, and use such characterization to classify the umbilic-free Euclidean hypersurfaces of dimension n ≥ 5 that admit non-trivial infinitesimal Moebius variations.

show abstract

Section: Proof Of Theoremmentioning

confidence: 90%

Section: Introductionmentioning

confidence: 99%

Infinitesimally Bonnet Bendable Hypersurfaces

Jimenez

Tojeiro

2023

J Geom Anal

View full text Add to dashboard Cite

show abstract

“…Therefore, the fact that and satisfy (22) implies that and A h also satisfy (22) with respect to . Since, in addition, , it follows from Proposition 8 of [11], together with Theorem 4.7 in [6] (also stated in [11] as Theorem 2), that h is locally infinitesimally Bonnet bendable, hence isothermic by Proposition 9 of [11].…”

Section: Proof Of Theoremmentioning

confidence: 96%

“…The study of hypersurfaces , , that admit non-trivial variations preserving the induced metric goes back to Sbrana [15] and Cartan [1] (see also [3] or Chapter 11 of [9]), whereas the hypersurfaces that admit non-trivial infinitesimal variations were investigated by Sbrana [14] (see also [10], Chapter 14 of [9] and [6]). We point out that the latter class turns out to be much larger than the former.…”

Section: Introductionmentioning

confidence: 99%

Infinitesimally Moebius bendable hypersurfaces

Jimenez,

Tojeiro

2024

Proceedings of the Edinburgh Mathematical Society

View full text Add to dashboard Cite

Li, Ma and Wang have provided in [13] a partial classification of the so-called Moebius deformable hypersurfaces, that is, the umbilic-free Euclidean hypersurfaces $f\colon M^n\to \mathbb{R}^{n+1}$ that admit non-trivial deformations preserving the Moebius metric. For $n\geq 5$ , the classification was completed by the authors in [12]. In this article we obtain an infinitesimal version of that classification. Namely, we introduce the notion of an infinitesimal Moebius variation of an umbilic-free immersion $f\colon M^n\to \mathbb{R}^m$ into Euclidean space as a one-parameter family of immersions $f_t\colon M^n\to \mathbb{R}^m$ , with $t\in (-\epsilon, \epsilon)$ and $f_0=f$ , such that the Moebius metrics determined by ft coincide up to the first order. Then we characterize isometric immersions $f\colon M^n\to \mathbb{R}^m$ of arbitrary codimension that admit a non-trivial infinitesimal Moebius variation among those that admit a non-trivial conformal infinitesimal variation, and use such characterization to classify the umbilic-free Euclidean hypersurfaces of dimension $n\geq 5$ that admit non-trivial infinitesimal Moebius variations.

show abstract

Infinitesimal variations of submanifolds

Cited by 2 publications

References 23 publications

Infinitesimally Bonnet Bendable Hypersurfaces

Infinitesimally Bonnet Bendable Hypersurfaces

Infinitesimally Moebius bendable hypersurfaces

Contact Info

Product

Resources

About