Parallel maps that preserve geometric objects of hypersurfaces
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.
scientific article; zbMATH DE number 3962675