Properties of C (1)-smooth functions whose gradient range has topological dimension 1

摘要

This paper continues the study of C1smooth func-tions of two variables whose gradient range has topo logical dimension 1 begun in [1–3]. The results findapplications in geometry (namely, in the theory ofdevelopable surfaces; see the first section in this paper) and in the theory of mappings with bounded distortion (see the second section in this paper). This study is also closely related to the problem of determining condi-tions on a set K under which the differential relation v ∈ K has nontrivial Lipschitz solutions (see, e.g., [4, 5]). We study a similar problem for C~1-smooth (not necessarily Lipschitz) solutions to differential rela- tions.