Abstract
This chapter summarizes the work of Calabi (Ann Math 58:1–23, 1953) about the existence of a Kahler immersion of a complex manifold into a finite or infinite dimensional complex space form. In particular, Calabi provides an algebraic criterion to find out whether a complex manifold admits or not such an immersion. Sections 2.1 and 2.2 are devoted to illustrate Calabi’s criterion for Kahler immersions into the complex Euclidean space and nonflat complex space forms respectively. In Sect. 2.3 we discuss the existence of a Kahler immersion of a complex space form into another, which Calabi himself in (Ann Math 58:1–23, 1953) completely classified as direct application of his criterion.
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