Umehara algebra and complex submanifolds of indefinite complex space forms

The Umehara algebra is studied with motivation on the problem of non-existence common complex submanifolds. In this paper, we prove some new results in and obtain applications. particular, if a manifolds admits holomorphic polynomial isometric immersion to one indefinite space form, then it cannot another form different type. Other consequences include submanifolds for projective or hyperbolic manifold distinguished metric, such as homogeneous domains, Hartogs triangle, minimal ball, symmetrized polydisc, equipped their intrinsic Bergman metrics, which generalizes more less all existing results.