On generalised Chen and \(k\)-minimal immersions (Q1356600)

Let \(f:M^m \to\mathbb{R}^{m+d}\) be a connected smooth immersion without boundary, \(N_x(f)\) be the normal space at a point \(x\), \(\{e_0, \dots, e_{d-1}\}\) be an orthonormal basis of \(N_x(f)\), \(H\) be the mean curvature and \(A\) the shape operator of \(M^m\). Define \({\mathcal A}_1(H) =\sum (\text{trace} A_H A_{e_i})e_i\) and \({\mathcal A}_k(H) =\sum (\text{trace} A_{{\mathcal A}_{k-1} (H)}A_{e_i}) e_i\). \(f\) is said to be an \({\mathcal A}_k\)-immersion if \({\mathcal A}_k(H) =0\). On the other hand, \textit{B. Rouxel} [Kodai Math. J. 4, 181-188 (1981; Zbl 0467.53004)] defined a linear map \(Q:N_x (f)\to N_x(f)\) by \(Q(u,v)= (\text{trace} A_u) (\text{trace} A_v)- \text{trace} (A_uA_v)\), considered the distinct eigenvalues \(\lambda_1, \dots, \lambda_s\) of \(Q\) with corresponding eigenspaces \(E_1, \dots, E_s\) so that \(N_x(f)= E_1 \oplus \cdots \oplus E_s\), and defined \(q(H)\) as the number of nonzero projections of \(H\) on the eigenspaces \(E_1, \dots,E_s\). Then it is known that \(q(H) =0\) if and only if \(H=0\) and \(q(H) =1\) if and only if \({\mathcal A}_1 (H)=0\). The authors of the present paper investigate the relation between \({\mathcal A}_k\)-submanifolds and the number \(q(H)\) for \(k\geq 1\). Furthermore, they introduce the ideal of a \(k\)-minimal submanifold which satisfies a stronger condition than that of an \({\mathcal A}_k\)-submanifold and give a method for constructing \(k\)-minimal submanifolds.