A complete solution of Samuel's problem

TL;DR

This paper provides a comprehensive solution to Samuel's problem in submanifold theory, characterizing pairs of immersions with identical Gauss maps and conformal induced metrics, extending previous work on isometric cases.

Contribution

It offers a complete classification of such immersion pairs, filling a gap left by earlier partial solutions and advancing understanding in submanifold geometry.

Findings

Characterization of all pairs of immersions with the same Gauss map and conformal metrics.

Extension of previous results from isometric to conformal cases.

Resolution of Samuel's problem posed in 1947.

Abstract

We give a complete solution of a problem in submanifold theory posed and partially solved by the eminent algebraic geometer Pierre Samuel in 1947. Namely, to determine all pairs of immersions of a given manifold into Euclidean space that have the same Gauss map and induce conformal metrics on the manifold. The case of isometric induced metrics was solved in 1985 by the first author and D. Gromoll.