Beitraege zur Algebra und Geometrie
Contributions to Algebra and Geometry
Volume 34 (1993), No. 2, 287-290 

Transnormal Graph Deformations

F. J. Craveiro de Carvalho

In [5] Stewart Robertson proved a
theorem which implies that every transnormal plane curve
can be deformed into a circle through transnormal curves.
At about the same time M.C. Irwin showed that under a certain
assumption transnormal embeddings of $S^1$ 
into $R^n$ are transnormally isotopic to spherical ones. The
assumption that Irwin needed was soon removed by Bernd Wegner
in [6]. This kind of results motivated the present work.

The curves we are dealing with are graphs of smooth maps defined
either on $R$ or $S^1$. We shall show that if the graph of such a
map is transnormal then it can be deformed through transnormal graphs
to the graph of a constant map. The proofs for $R$ and
$S^1$ are similar but for the sake of the exposition we
shall treat the two cases separately.