Parallel maps that preserve geometric objects of hypersurfaces
type: Item
instance of: scholarly article
title: Parallel maps that preserve geometric objects of hypersurfaces
zbMATH Open document ID: 0597.53046
DOI: 10.1216/RMJ-1986-16-1-103
author: Krishna Amur
published in: Rocky Mountain Journal of Mathematics
publication date: 1986-01-01T00:00:00Z
review text: Let M be a connected, orientable smooth manifold of dimension n and \(X: M\to {\mathbb{R}}^{n+1}\) an immersion. The immersions \(X_ t\), \(X_{-t}: M\to R^{n+1}\), given by \(X_ t(p)=X(p)+t\cdot N(X(p))\) and \(X_{- t}=X(p)-t\cdot N(X(p))\) with t sufficiently small and \(p\in M\), lead to the definition of \(f_ t: X(M)\to X_ t(M)\) and \(f_{-t}: X(M)\to X_{-t}(M)\), which are assumed to be local diffeomorphisms. It is known that the parallel map \(f_ t^{\#}=f_ t\circ f^{-1}_{-t}\) preserves principal directions, umbilics and the third fundamental form [cf. \textit{R. B. Gardner}, Proc. Sympos. Pure Math. 15, 231-237 (1970; Zbl 0206.502)]. The present author proves some additional properties of \(f_ t^{\#}\). Let C(t) be the Jacobian matrix of \(f_ t^{\#}:\) (a) If the principal curvatures \(k_ i\) of M are all non-zero, det C(t)\(=1\) for all such t (i.e. C(t)\(\in SL(n,{\mathbb{R}}))\) and \(n=2m\), then X(M) is a nontrivial immersed minimal hypersurface of \({\mathbb{R}}^{2m+1}\) and \(f_ t^{\#}\) preserves (i) the element of area, (ii) all even order elementary symmetric functions of principal curvatures and (iii) the absolute value of each odd order elementary symmetric function of principal curvatures. (b) If \(C(t)=\lambda I\), for some \(\lambda >0\), then X(M) is an umbilical hypersurface or a hyperplane in \({\mathbb{R}}^{n+1}\) and \(f_ t^{\#}\) is conformal. In particular, (i) if \(\lambda\) \(\neq 1\) and X(M) is compact, then X(M) is a Euclidean sphere and (ii) if \(\lambda =1\), then X(M) is a hyperplane in \({\mathbb{R}}^{n+1}\) and \(f_ t^{\#}\) is an isometry.
Mathematics Subject Classification ID: 53C40
Mathematics Subject Classification ID: 53A07
zbMATH DE Number: 3962675
zbMATH Keywords: parallel surfaces
zbMATH Keywords: parallel map
zbMATH Keywords: principal curvatures
zbMATH Keywords: minimal hypersurface
zbMATH Keywords: elementary symmetric functions
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3962675