On minimal submanifolds in product manifolds with a certain Riemannian metric (Q1375735)

The author generalizes \textit{N. Ejiri}'s theorem on minimal submanifolds in warped product manifolds [Trans. Am. Math. Soc. 276, 347-360 (1983; Zbl 0511.53063)]. Let \((N_1,g_1), \dots, (N_l,g_l)\) be Riemannian manifolds and \(\dim N_i=n_i\), \(i=1, \dots,l\). Denote by \({\mathcal M}'\) the set of all Riemannian metrics on the product manifold \(N=N_1 \times \cdots \times N_l\). A subset \({\mathcal M}\) of \({\mathcal M}'\) is defined to be \({\mathcal M}= \{g\in {\mathcal M}': g=f^2_1f_1 +\cdots +f^2_l g_l\}\), where \(f_1, \dots, f_l\) are positive smooth functions on \(N\). Such an element \(g=\sum f^2_ig_i\) may be abbreviated as \((f_1, \dots, f_l)\). Let \(d_1, \dots, d_l\) be positive integers. Two elements \(g=(f_1, \dots, f_l)\) and \(\overline g=(\overline f_1, \dots, \overline f_l)\) are said to be \((d_1, \dots, d_l)\)-equivalent, denoted by \(g\sim_{(d_1, \dots, d_l)} \overline g\), if \(f^{d_1}_1 \cdots f^{d_l}_l= \overline f_1^{d_1} \cdots \overline f_l^{d_l}\) holds. Let \(\varphi_i: M_i\to N_i\) be an immersion of a \(d_i\)-dimensional manifold \(M_i\) into \(N_i\), \(i=1, \dots,l\), and let \(\Phi: M\to N\) be the product immersion where \(M=M_1 \times \cdots \times M_l\). In present paper, the following results are shown. (1) Assume that \(g\sim_{(d_1, \dots, d_l)} \overline g\). Then \(\Phi: M\to (N,g)\) is minimal if and only if \(\Phi: M \to (N, \overline g)\) is minimal. (2) Let \(F_i\) be a positive smooth function on \(N_i\), \(i=1, \dots, l\). Assume that \(g\sim_{(d_1, \dots, d_l)} (F_1, \dots, F_l)\). Then \(\Phi: M\to (N,g)\) is minimal if and only if each \(\varphi_i: M_i\to (N_i,F^2_i g_i)\) is minimal, \(i=1, \dots,l\).