Infinitesimally Bonnet Bendable Hypersurfaces

Abstract

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.

Infinitesimally Moebius bendable hypersurfaces

“…But since rank ker Ā = n − 1 on V 2 , then h has index of relative nullity equal to one at any point. By Corollary 12 in [11], h is a generalized cone over a unit-speed curve γ : J → Q 2 c in an umbilical surface Q 2 c ⊂ Q 3 , c ≥ . Cases (iii) and (iv): Let W be a connected component of V 1 ∩ W 0 2 ∩ U 0 2 (respectively, V 1 ∩ W 0 2 ∩ Y 0 1 ).…”

“…For the proofs of the next two lemmas we refer to [11] (see Lemma 4 and Lemma 5 therein, respectively).…”

“…Notice that any surface as in (ii ), (iii ) and (iv ) is also isothermic (for h as in (ii ) see Corollary 12 in [11]). Notice also that f | W being a conformally surface-like hypersurface determined by a surface h :…”

“…Notice that any surface as in ( ii ), ( iii ) and ( iv ) is also isothermic (for h as in ( ii ) see Corollary 12 in [11]). Notice also that being a conformally surface-like hypersurface determined by a surface , , is equivalent to being (isometric to) a Riemannian product and to being given by , where i is the inclusion map of an open subset of either or , according to whether ϵ is zero or not, is a Moebius transformation of , and Φ is the standard isometry if ϵ = 0 and, if or 1, respectively, the conformal diffeomorphism:…”

“…The associated variational vector fields are called infinitesimal bendings and correspond to the conformal infinitesimal bendings with conformal factor ρ = 0. The reason why isothermic surfaces appear in this context is that they are precisely the surfaces that are locally infinitesimally Bonnet bendable , that is, the surfaces that admit local infinitesimal variations such that the mean curvature functions H t of , , coincide up to the first order, that is, (see, e.g., Proposition 9 of [11]).…”