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On isometric minimal immersions from warped products into real space forms. (English) Zbl 1022.53022

The product of two manifolds \(N_1\) and \(N_2\) with Riemannian metrics \(g_{N_1}\) and \(g_{N_2}\) is called a warped product \(N_1\times_f N_2\) if for a certain positive “warping function” \(f\) on \(N_1\) this product is equipped with the Riemannian metric \(g:= g_{N_1}+ f^2 g_{N_2}\). The author handles the problem to discover necessary conditions for a given warped product to admit a minimal isometric immersion \(\phi\) into a \(m\)-dimensional space form \(R_m(c)\) of constant curvature \(c\). For this purpose he uses the fundamental inequality \[ {\Delta f\over f}\leq{(n_1+ n_2)^2\over 4n_2} H^2+ n_1c \] (\(H^2=\) squared mean curvature vector of \(\phi\), \(\Delta=\) Laplacian of \(N_1\), \(n_i=\dim N_i\) for \(i= 1,2\)) for any isometric immersion into \(R_m(c)\). This inequality yields some theorems about the nonexistence of minimal isometric immersions \((H^2= 0)\) of the warped product into the hyperbolic or the Euclidean space. Certain examples show that these results are best possible.

MSC:

53B25 Local submanifolds
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53C40 Global submanifolds
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