Normal curvature of CR submanifolds of maximal CR dimension of the complex projective space
If \((\overline M,J)\) is a complex manifold of dimension \(\frac{n+p}{2}\) equipped with a Hermitian metric \(\overline g\), then an \(n\)-dimensional submanifold \(M\) is CR submanifold of maximal CR dimension if at each point \(x\in M\) the real dimension of \(JT_x(M)\cap T_x(M)\) is \(n-1\). The submanifold \(M\) is odd-dimensional and there exists a unit vector field \(\xi\) normal to \(TM\) such that \(JT_x(M)\subset T_x(M)\oplus\text{span}\{\xi_x\}\). If the normal curvature of \(M\) in \(\overline M\) vanishes identically, then it is said that the normal connection of \(M\) is flat. In this paper, the authors prove that there do not exist CR submanifolds \(M\) of maximal CR dimension of a complex projective space \(\mathbf{P(C)}\) of dimension \(\frac{n+p}{2}\) with flat normal connection \(D\) of \(M\) when the distinguished normal vector field \(\xi\) is parallel with respect to \(D\). For a submanifold \(M\) of \(\mathbf{P(C)}\), it is well known that \(\pi^{-1}(M)\) is a submanifold of \(\mathbf{S}^{n+p+1}\), where \(\pi^{-1}(M)\) is the circle bundle over \(M\)which is compatible with the Hopf map \(\pi:\mathbf{S}^{n+p+1}\to\mathbf{P(C)}\). If the normal connection of \(\pi^{-1}(M)\) in \(\mathbf{S}^{n+p+1}\) is flat, it is said that the normal connection of \(M\) is lift-flat. The authors also prove that if \(M\) is a CR submanifold of maximal CR dimension of a complex projective space \(\mathbf{P(C)}\) with lift-flat normal connection \(D\) and \(\xi\) is parallel with respect to \(D\), then there exists a totally geodesic complex projective subspace \(\mathbf{P}^{\frac{n+1}{2}}\mathbf{(C)}\) of \(\mathbf{P(C)}\) such that \(M\) is a real hypersurface of \(\mathbf{P}^{\frac{n+1}{2}}\mathbf{(C)}\).
scientific article; zbMATH DE number 7011025