A characterization of constant isotropic immersions by circles
Motivated by the well-known result of Nomizu and Yano on circles and spheres in Riemannian geometry, the author provides a characterization of constant isotropic immersions into an arbitrary Riemannian manifold by circles on the submanifolds. Let \(M^n\) be an \(n\)-dimensional connected Riemannian submanifold of an \((n + p)\)-dimensional Riemannian manifold \(\widetilde M^{n+p}\) through an isometric immersion \(f\). Then the following are equivalent: (i) \(M^n\) is a constant (\(\lambda\)-) isotropic submanifold of \(\widetilde M^{n+p}\). (ii) There exists \(\kappa > 0\) such that for each circle \(\gamma\) of curvature \(\kappa\) on the submanifold \(M^n\) the curve \(f\circ\gamma\) in \(\widetilde M^{n+p}\) has constant first curvature \(\kappa_1\) along this curve. As an immediate consequence of this result, the author characterizes Veronese imbeddings of complex projective spaces into complex projective spaces which are typical examples of Kähler immersions. Let \(f : M_n\to M_N(c)\) be a Kähler isometric full immersion of an \(n\)-dimensional Kähler manifold \(M_n\) into an \(N\)-dimensional complex space form \(M_N(c)\) of constant holomorphic sectional curvature \(c > 0\). Then the following conditions are equivalent. (i) For some positive integer \(k\), the submanifold \(M_n\) is locally congruent to \(\mathbb CP^n( c/k)\), \(N=\binom{n+k}{n}-1\) and \(f\) is locally equivalent to the \(k\)-th Veronese imbedding \(f_k\). (ii) There exists \(\kappa > 0\) such that for each circle \(\gamma\) of curvature \(\kappa\) on the submanifold \(M_n\) the curve \(f\circ\gamma\) in \(M_N(c)\) has constant first curvature \(\kappa_1\) along this curve.
Veronese imbeddings of complex projective spaces
scientific article; zbMATH DE number 2051580