Infinitesimally Bonnet Bendable Hypersurfaces

Author(s)

Jimenez, M. I. ; Tojeiro, R.

• Document type: Journal article
• Source: JOURNAL OF GEOMETRIC ANALYSIS; v. 33, n. 5, p. 17-pg., 2023-05-01.

Abstract

The classical Bonnet problem is to classify all immersions f : M-2 -> R-3 into Euclidean three-space that are not determined, up to a rigid motion, by their induced metric and mean curvature function. The natural extension of Bonnet problem for Euclidean hypersurfaces of dimension n >= 3 was studied by Kokubu (Tohoku Math J 44:433-442, 1992). In this article, we investigate an infinitesimal version of Bonnet problem for hypersurfaces with dimension n >= 3 of any space form, namely, we classify the hypersurfaces f :Mn -> Qnc +1 , n >= 3, of any space form Qn+1 c of constant curvature c, for which there exists a (non-trivial) one-parameter family of immersions f(t) : M-n -> Q(c)(n+1) , with f(0) = f , whose induced metrics gt and mean curvature functions H-t coincide up to the first order, " that is, & part;/& part;t|t=0gt =0 = & part;/& part;t|H-t=0(t). (AU)

• FAPESP's process: 16/23746-6 - Algebraic, topological and analytical techniques in differential geometry and geometric analysis
• Grantee: Paolo Piccione
• Support Opportunities: Research Projects - Thematic Grants