Calabi’s Criterion

  • Andrea Loi
  • Michela Zedda
Part of the Lecture Notes of the Unione Matematica Italiana book series (UMILN, volume 23)


This chapter summarizes the work of Calabi (Ann Math 58:1–23, 1953) about the existence of a Kähler 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 Kähler immersions into the complex Euclidean space and nonflat complex space forms respectively. In Sect. 2.3 we discuss the existence of a Kähler 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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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Andrea Loi
    • 1
  • Michela Zedda
    • 2
  1. 1.Department of Mathematics & Computer ScienceUniversity of CagliariCagliariItaly
  2. 2.Department of Mathematical, Physical & Computer SciencesUniversity of ParmaParmaItaly

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