Differential geometry of submanifolds of warped product manifolds \(I \times _f S^{m-1} (k)\)
At first some basic results are given for general submanifolds in the Riemannian warped product \(R^m(k,f)=(I\times S,g^k_f)\), \(g^k_f= dt^2+f^2(t)g_k\), where \(f\) is a positive function on \(I\) and \((S,g_k)\) is a Riemannian 3-manifold of constant sectional curvature \(k\). A submanifold \(M\) of \(R^m(k,f)\) is called transverse if it is contained in a slice \(S(t_0):=(t_0)\times S\), and horizontal if the horizontal vector field \(\partial_t\) is tangent to \(M\) at each point on \(M\). A submanifold \(M\) of a Riemannian manifold \(\overline M\) is called curvature-invariant if \(\overline R(X,Y)(T_pM)\subset T_pM\) holds for \(X,Y\in T_pM,\) \(p\in M\). It is proved that if \(R^m(k,f)\) contains no open subsets of constant curvature, then a submanifold \(M\) in \(R^m(k,f)\) is curvature-invariant if and only if either \(M\) is a transverse submanifold or an horizontal submanifold; these are described in more details. Also the minimal submanifolds and their optimal curvature estimates are studied. In the following sections in \(R^m(k,f)\) the parallel submanifolds and the hypersurfaces of constant sectional curvature are classified, and the totally umbilical submanifolds are investigated.
scientific article; zbMATH DE number 5562622