Warped products in real space forms
Let \(B\) and \(F\) be two Riemannian manifolds with metric \(g_B\) and \(g_F\), respectively, and \(f\neq0\) be a function on \(B\). For the product manifold \(B\times F\) with projection \(\pi: B\times F\to B\), \(\eta: B\times F\to F\), the warped product \(M= B\times_fF\) is the manifold \(B\times F\) equipped with the metric such that \[ \| X\|^2= \|\pi_*(X)\|^2+ f^2(\pi(x))\|\eta_*(X)\|^2 \] for any \(X\in T_x(M)\) so that \(g= g_B+ f^2 g_F\). For a submanifold \(N\) in a Riemannian manifold \(\overline M\) one denotes by \(\nabla\) and \(\overline\nabla\) the Levi-Cività connections of \(N\) and \(\overline M\), respectively. Then the Gauss equation at \(p\in N\) is \(\overline\nabla_X Y= \nabla_X Y+ h(X,Y)\) for \(X,Y\in T_pN\). The mean curvature vector \(\widetilde H\) is given by \(\overline H= {1\over n}\sum^n_{i=1} h(e_i, e_i)\), where \(\{e_1,\dots, e_n\}\) is an orthonormal frame of \(TN\). Now, assume that \(\phi: N_1\times_f N_2\to R^m(c)\) is an isometric immersion of a warped product \(N_1\times_f N_2\) into a Riemannian manifold of constant sectional curvature \(c\). Then, the following theorem holds: (1) The scalar curvature \(\tau\) of \(N_1\times_f N_2\) satisfies \[ \tau\leq {\Delta f\over n_1f}+ {n^2(n- 2)\over 2(n-1)} H^2+{1\over 2} (n+ 1)(n- 2)c, \] where \(n_1= \dim N_1\), \(n= \dim N_1\times N_2\). \(H^2\) is the squared mean curvature of \(\phi\) and \(\Delta\) is the Laplacian of \(N_1\). (2) If \(n\geq 2\), the equality sign holds automatically. (3) If \(n\geq 3\), the equality sign holds if and only if one of the following two cases occurs: (3a) \(N_1\times_f N_2\) is of constant sectional curvature \(c\) where the warping function \(f\) satisfies \(\Delta f= cf\) and \(N_1\times_f N_2\) is totally geodesic in \(R^m(c)\). (3) \(N_1\times_f N_2\) can be locally immersed as a rotational hypersurface in a totally geodesic submanifold \(R^{n+1}(c)\) of \(R^m(c)\), if \(H^2\) is positive.
scientific article; zbMATH DE number 2104650