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**Distinguished connections on \((J^2=\pm 1)\)-metric manifolds.**
*(English)*
Zbl 1389.53052

A systematical exposition of the theory of reducible connections on a \((J^2=\pm 1)\)-metric manifold is the subject of this nice paper.

A \((J^2=\pm 1)\)-metric manifold is a manifold endowed with an \((\alpha ,\varepsilon)\)-structure \((J,g)\), \(\alpha\), \(\varepsilon\) belonging to \(\{-1,1\}\), \(J\) being a \((1,1)\)-tensor field such that \(J^2=\alpha\mathrm{Id}\), and \(g\) a \((0,2)\)-tensor field such that \(g(JX,JY)=\varepsilon g(X,Y)\), for any vector fields \(X,Y\). Thus, there exist four kinds of \((\alpha ,\varepsilon)\)-structures, according to the values of \(\alpha\) and \(\varepsilon\). This allows to consider the following types of \((J^2=\pm 1)\)-metric manifolds \((M,J,g)\).

Firstly, the authors characterize reducible connections on a \((J^2=\pm 1)\)-manifold. They define two adapted connections which generate a \(1\)-parameter family of adapted connections. Then, they present Kobayashi-Nomizu and Yano-type connections. This is the subject of Section \(2\).

Section \(3\) is devoted to the study of reducible connections on a \((J^2=\pm 1)\)-metric manifold. In particular, the existence and uniqueness of a reducible connection, called the first canonical connection, is stated. This allows to specify the set of canonical connections considering the first canonical connection as a starting point.

Section \(4\), which has technical character, is devoted to the study of three tensors: the covariant derivative of the fundamental tensor with respect to the Levi-Civita conection, the Nijenhuis tensor \(N_J\) and the second Nijenhuis tensor. In particular, the authors prove that, if \(\alpha\varepsilon =1\), the second Nijenhuis tensor vanishes if and only if the manifold is of quasi-Kähler type. This resembles a similar result which involves the Nijenhuis tensor if \(\alpha\varepsilon =-1\).

Sections \(5\), \(6\) are the core of the paper. Firstly, the authors study three natural connections: the first canonical connection \(\nabla^0\), the Chern and the well-adapted connection \(\nabla^w\). The Chern connection, which can be defined only when \(\alpha\varepsilon =-1\), coincides with \(\nabla^0\) if and only if \(J\) is integrable. Moreover, if \(\alpha\varepsilon =-1\), then \(\nabla^0\) and \(\nabla^w\) coincide if and only if the \((\alpha ,\varepsilon)\)-structure is of quasi-Kähler type.

On any \((J^2=\pm 1)\)-metric manifold, the Levi-Civita, the Kobayashi-Nomizu and the Yano connections can be defined, but they are not natural, in general. So, the authors state conditions for their naturality. Moreover, they characterize the existence of an adapted connection with skew-symmetric torsion. This characterization, obtained in a unitary way, involves the behaviour of the Nijenhuis tensor and is used for a new proof as a known result. In particular, if \(\alpha\varepsilon =1\), a natural connection with skew-symmetric torsion does exist if and only if the structure is of quasi-Kähler type.

Canonical connections are studied in Section 6. If \(\alpha\varepsilon =-1\), they set up a \(1\)-parameter family of adapted connections, which is the affine line determined by the first canonical and the Chern connections. Since \(\nabla^w\) belongs to this line, the family is regarded as the line determined by \(\nabla^0\) and \(\nabla^w\). This allows to define canonical connections in the case \(\alpha\varepsilon =1\), also. In fact, the covariant derivative of a canonical connection on any \((J^2=\pm 1)\)-metric manifold acts as: \(\nabla_X^SY=(1-s)\nabla_X^0Y+s\nabla_X^wY\), \(s\) being a real number. Moreover, if \(\alpha\varepsilon =1\), the authors obtain another parametrization for the family, which involves \(\nabla^0\) and \(N_J\). Note that, if \(\alpha\varepsilon =-1\), canonical connections can be parametrized by means of \(\nabla^0\) and the second Nijenhuis tensor.

A \((J^2=\pm 1)\)-metric manifold is a manifold endowed with an \((\alpha ,\varepsilon)\)-structure \((J,g)\), \(\alpha\), \(\varepsilon\) belonging to \(\{-1,1\}\), \(J\) being a \((1,1)\)-tensor field such that \(J^2=\alpha\mathrm{Id}\), and \(g\) a \((0,2)\)-tensor field such that \(g(JX,JY)=\varepsilon g(X,Y)\), for any vector fields \(X,Y\). Thus, there exist four kinds of \((\alpha ,\varepsilon)\)-structures, according to the values of \(\alpha\) and \(\varepsilon\). This allows to consider the following types of \((J^2=\pm 1)\)-metric manifolds \((M,J,g)\).

- i)
- Almost Hermitian manifold, namely \((\alpha ,\varepsilon)=(-1,1)\) and \(g\) is a Riemannian metric,
- ii)
- Almost Norden manifold, namely \((\alpha ,\varepsilon)=(-1,-1)\), \(g\) has signature \((n,n)\), with \(\dim M=2n\),
- iii)
- Almost product Riemannian manifold, that is \((\alpha ,\varepsilon)=(1,1)\), \(g\) is a Riemannian metric and assume that the trace of \(J\) vanishes,
- iv)
- Almost para-Hermitian manifold, that is \((\alpha ,\varepsilon)=(1,-1)\) and \(g\) is a semi-Riemannian metric of signature \((n,n)\).

Firstly, the authors characterize reducible connections on a \((J^2=\pm 1)\)-manifold. They define two adapted connections which generate a \(1\)-parameter family of adapted connections. Then, they present Kobayashi-Nomizu and Yano-type connections. This is the subject of Section \(2\).

Section \(3\) is devoted to the study of reducible connections on a \((J^2=\pm 1)\)-metric manifold. In particular, the existence and uniqueness of a reducible connection, called the first canonical connection, is stated. This allows to specify the set of canonical connections considering the first canonical connection as a starting point.

Section \(4\), which has technical character, is devoted to the study of three tensors: the covariant derivative of the fundamental tensor with respect to the Levi-Civita conection, the Nijenhuis tensor \(N_J\) and the second Nijenhuis tensor. In particular, the authors prove that, if \(\alpha\varepsilon =1\), the second Nijenhuis tensor vanishes if and only if the manifold is of quasi-Kähler type. This resembles a similar result which involves the Nijenhuis tensor if \(\alpha\varepsilon =-1\).

Sections \(5\), \(6\) are the core of the paper. Firstly, the authors study three natural connections: the first canonical connection \(\nabla^0\), the Chern and the well-adapted connection \(\nabla^w\). The Chern connection, which can be defined only when \(\alpha\varepsilon =-1\), coincides with \(\nabla^0\) if and only if \(J\) is integrable. Moreover, if \(\alpha\varepsilon =-1\), then \(\nabla^0\) and \(\nabla^w\) coincide if and only if the \((\alpha ,\varepsilon)\)-structure is of quasi-Kähler type.

On any \((J^2=\pm 1)\)-metric manifold, the Levi-Civita, the Kobayashi-Nomizu and the Yano connections can be defined, but they are not natural, in general. So, the authors state conditions for their naturality. Moreover, they characterize the existence of an adapted connection with skew-symmetric torsion. This characterization, obtained in a unitary way, involves the behaviour of the Nijenhuis tensor and is used for a new proof as a known result. In particular, if \(\alpha\varepsilon =1\), a natural connection with skew-symmetric torsion does exist if and only if the structure is of quasi-Kähler type.

Canonical connections are studied in Section 6. If \(\alpha\varepsilon =-1\), they set up a \(1\)-parameter family of adapted connections, which is the affine line determined by the first canonical and the Chern connections. Since \(\nabla^w\) belongs to this line, the family is regarded as the line determined by \(\nabla^0\) and \(\nabla^w\). This allows to define canonical connections in the case \(\alpha\varepsilon =1\), also. In fact, the covariant derivative of a canonical connection on any \((J^2=\pm 1)\)-metric manifold acts as: \(\nabla_X^SY=(1-s)\nabla_X^0Y+s\nabla_X^wY\), \(s\) being a real number. Moreover, if \(\alpha\varepsilon =1\), the authors obtain another parametrization for the family, which involves \(\nabla^0\) and \(N_J\). Note that, if \(\alpha\varepsilon =-1\), canonical connections can be parametrized by means of \(\nabla^0\) and the second Nijenhuis tensor.

Reviewer: Maria Falcitelli (Bari)

### MSC:

53C15 | General geometric structures on manifolds (almost complex, almost product structures, etc.) |

53C05 | Connections (general theory) |

53C07 | Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills) |

53C50 | Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics |