Totally real submanifolds of \(\mathbb{C} P^ n\) satisfying a basic equality
Let \(M^ n\) be an \(n\)-dimensional totally real submanifold of an \(m\)- dimensional complex space form \(\widetilde M^ m(4c)\) with constant holomorphic sectional curvature \(4c\). We define a Riemannian invariant \(\delta_ m\) of \(M^ n\) by \(\delta_ M(p) = \tau(p) - \inf K(p)\), where \(\inf K\) is the function assigning to each \(p \in M^ n\) the infimum of \(K(\pi)\), where \(\pi\) runs over all planes in \(T_ pM\) and \(\tau\) is the scalar curvature, defined by \(\tau = \sum_{i < j} K(e_ i \wedge e_ j)\). Then the following inequality holds: \[ \delta_ M \leq {n^ 2(n-2) \over 2(n-1)} \| H\|^ 2 + {1\over 2}(n + 1)(n + 2)c \] where \(H\) is the mean curvature vector. In this paper we study totally real submanifolds which realize the equality in this inequality. Then we say that \(M\) satisfies Chen's equality. First we prove that, if \(M\) satisfies Chen's equality, then \(H(p)\) is orthogonal to \(J(T_ p M)\) at any point \(p\), where \(J\) denotes the complex structure of the complex space form. Hence any Lagrangian submanifold (\(n = m\)) which satisfies Chen's equality automatically is minimal. Next we study \(n\)-dimensional totally real submanifolds in \(\mathbb{C} P^ n(4)\) satisfying Chen's equality. We define for every \(p \in M\) the kernel of the second fundamental form by \[ {\mathcal D}(p) = \{X \in T_ p M \mid \forall Y \in T_ p M: h(X,Y) = 0\}. \] If the dimension of \({\mathcal D}(p)\) is constant, then it follows that either \(M\) is totally geodesic or that the distribution \(\mathcal D\) is an \((n-2)\)-dimensional completely integrable distribution. We give a complete classification of all 3- dimensional totally real submanifolds of \(\mathbb{C} P^ 3\) satisfying the dimension of \(\mathcal D\) is constant (and hence it is a completely integrable distribution), the distribution \({\mathcal D}^ \perp\) is also integrable. For the classification, we make use of the notion of warped product immersions, as introduced by S. Noelker.
scientific article; zbMATH DE number 734498