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Almost complex manifolds with non-degenerate torsion

Cristina Bozzetti

Dipartimento di Matematica ed Applicazioni, Università di Milano-Bicocca, Via Cozzi 55, 20125 Milano, Italy

E-mail Address: cristina.bozzetti.83@gmail.com

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and

Costantino Medori

Dipartimento di Matematica e Informatica, Università di Parma, Parco Area Delle Scienze 53/A, 43124 Parma, Italy

E-mail Address: costantino.medori@unipr.it

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https://doi.org/10.1142/S0219887817500335Cited by:0 (Source: Crossref)

Abstract

We show that almost complex manifolds $(M^{4}, J)$ of real dimension 4 for which the image of the Nijenhuis tensor forms a non-integrable bundle, called torsion bundle, admit a $ℤ_{2}$ -structure locally, that is, a double absolute parallelism. In this way, the problem of equivalence for such almost complex manifolds can be solved; moreover, the classification of locally homogeneous manifold $(M^{4}, J)$ is explicitly given when the Lie algebra of its infinitesimal automorphisms is non-solvable (indeed reductive). It is also shown that the group of the automorphisms of $(M^{4}, J)$ is a Lie group of dimension less than or equal to 4, whose isotropy subgroup has at most two elements, and that there are not non-constant holomorphic functions on $(M^{4}, J)$ .

Keywords:

AMSC: 32Q60, 53C15, 57S20

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