Communications in Analysis and Geometry

Volume 25 (2017)

Number 4

Classification of codimension two deformations of rank two Riemannian manifolds

Pages: 751 – 797



Luis A. Florit (Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, Brazil)

Guilherme M. de Freitas (Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, Brazil)


The purpose of this work is to close the local deformation problem of rank two Euclidean submanifolds in codimension two by describing their moduli space of deformations. In the process, we provide an explicit simple representation of these submanifolds, a result of independent interest by its applications. We also determine which deformations are genuine and honest, allowing us to find the first known examples of honestly locally deformable rank two submanifolds in codimension two. In addition, we study which of these submanifolds admit isometric immersions as Euclidean hypersurfaces, a property that gives rise to several applications to the Sbrana–Cartan theory of deformable Euclidean hypersurfaces.

Received 2 October 2015

Published 1 November 2017