Certain condition on the second fundamental form of CR submanifolds of maximal CR dimension of complex projective space
Let \((\overline M, g, J)\) be a Kähler manifold and \(M \subset \overline M\) a real submanifold. For any \(x \in M\) denote by \(\mathcal D_x\) the maximal \(J\)-invariant subspace \(\mathcal D_x \subset T_x M\). We recall that, when \(\dim \mathcal D_x\) is constant, the submanifold \(M\) is called \textit{CR manifold} and that, when \(\dim \mathcal D_x = \dim M - 1\), \(M\) is called \textit{of maximal CR dimension}. In this case, for any \(x \in M\) there exists a unit vector \(\xi_x \in (T_x M)^\perp\) so that \(J(T_x M) = \mathcal D_x + <J \xi_x, \xi_x> \) and there exists a tensor field \(F\) on \(M\), of type \((1,1)\), which is defined for any \(x \in M\) and \(v \in T_x M\) by the relation \(J v = F_x (v) - g(v, J\xi_x) \xi_x\). In this paper the authors consider the submanifolds of maximal CR dimension in complex projective spaces and obtain the following main result: Let \(M\) be a complete \(m\)-dimensional submanifold of maximal CR dimension of \(\mathbb C P^{\frac{m+p}{2}}\) and \(h\) the second fundamental form of \(M\). If for any vector field \(X\) and \(Y\) \[ h(F(X), Y) - h(X, F(Y)) = g(F(X), Y) \eta \;, \qquad \eta \in TM^\perp\;, \] then \(\mathcal D\) is a contact distribution and \(M\) is either congruent to a geodesic sphere or to a tube over a complex quadric or it is an invariant submanifold of a geodesic hypersphere.
CR submanifolds of complex projective spaces
scientific article; zbMATH DE number 5508733
scientific article