An exotic totally real minimal immersion of S³ in ℂP³ and its characterisation

In a previous paper, B.-Y. Chen defined a Riemannian invariant δ by subtracting from the scalar curvature at every point of a Riemannian manifold the smallest sectional curvature at that point, and proved, for a submanifold of a real space form, a sharp inequality between δ and the mean curvature function. In this paper, we extend this inequality to totally real submanifolds of a complex space form. As a consequence, we obtain a metric obstruction for a Riemannian manifold M n to admit a minimal totally real (i.e. Lagrangian) immersion into a complex space form of complex dimension n . Next we investigate three-dimensional submanifolds of the complex projective space ℂ P 3 which realise the equality in the inequality mentioned above. In particular, we construct and characterise a totally real minimal immersion of S 3 in ℂ P 3 .