Tohoku Mathematical Journal
2005
December
SECOND SERIES VOL. 57, NO. 4

Tohoku Math. J.
57 (2005), 469-503

Title ISOLATED ROUNDINGS AND FLATTENINGS OF SUBMANIFOLDS IN EUCLIDEAN SPACES

Author Toshizumi Fukui and Juan J. Nuño-Ballesteros

(Received May 30, 2003, revised May 11, 2005)
Abstract. We introduce the concepts of rounding and flattening of a smooth map $g$ of an $m$-dimensional manifold $M$ to the euclidean space $\boldsymbol{R}^n$ with $m<n$, as those points in $M$ such that the image $g(M)$ has contact of type $\Sigma^{m,\dots,m}$ with a hypersphere or a hyperplane of $\boldsymbol{R}^n$, respectively. This includes several known special points such as vertices or flattenings of a curve in $\boldsymbol{R}^n$, umbilics of a surface in $\boldsymbol{R}^3$, or inflections of a surface in $\boldsymbol{R}^4$.

2000 Mathematics Subject Classification. Primary 53A07; Secondary 58K05, 53A05.
Key words and phrases. Distance squared function, height function.

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