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Submanifolds with a non-degenerate parallel normal vector field in Euclidean spaces. (English) Zbl 1133.53001
Izumiya, Shyuichi (ed.) et al., Singularity theory and its applications. Papers from the 12th MSJ International Research Institute of the Mathematical Society of Japan, Sapporo, Japan, September 16–25, 2003. Tokyo: Mathematical Society of Japan (ISBN 978-4-931469-32-7/hbk). Advanced Studies in Pure Mathematics 43, 311-332 (2006).
The author considers the class of submanifolds $$M$$ in a Euclidean space $$\mathbb R^n$$ which admit a non-degenerate parallel normal vector field $$\nu$$. The image of the associated Gauss map $$G_{\nu}:M\rightarrow S^{n-1}$$ defines an immersed hyperspherical submanifold $$M^{\nu}$$ which has the following property: if $$M$$ has a contact of Boardman type $$\sum^{i_1,\dots,i_k}$$ with a hyperplane, then $$M^{\nu}$$ has the same contact type with the translated hyperplane. In particular, for a space curve $$\alpha$$ in $$\mathbb R^3$$, the spherical curve $$\alpha^{\nu}$$ has the same flattenings and an extension of the four vertex theorem. For an immersed surface $$M$$ in $$\mathbb R^4$$, it admits a local non-degenerate parallel normal vector field if and only if it is totally semi-umbilic and has non zero Gaussian curvature. Moreover, $$G_{\nu}$$ preserves the inflections and the asymptotic lines between $$M$$ and $$M^{\nu}$$. As a consequence, the author deduces an extension for this class of surfaces of the classical Loewner and Carathéodory conjectures for umbilic points of analytic immersed surfaces in $$\mathbb R^3$$.
For the entire collection see [Zbl 1110.32001].

##### MSC:
 53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related $$n$$-spaces