# The canonical involution in the space of connections of a $(J^{2}=\pm 1)$-metric manifold

## TL;DR

This paper introduces a canonical involution in the space of connections on $(J^{2}= ext{±}1)$-metric manifolds, linking various special connections and preserving metric properties.

## Contribution

It defines a canonical involution that projects the set of connections onto those adapted to the structure J, unifying several important connections in the geometry of such manifolds.

## Key findings

- The involution maps the Levi Civita connection to the first canonical connection.
- In the almost Hermitian case, it maps the
abla^{-} connection to the Chern connection.
- It preserves metric connections within the space of all connections.

## Abstract

A $(J^{2}=\pm 1)$-metric manifold has an almost complex or almost product structure $J$ and a compatible metric $g$. We show that there exists a canonical involution in the set of connections on such a manifold, which allows to define a projection over the set of connections adapted to $J$. This projection sends the Levi Civita connection onto the first canonical connection. In the almost Hermitian case, it also sends the $\nabla^{-}$ connection onto the Chern connection, thus applying the line of metric connections defined by $\nabla ^{-}$ and the Levi Civita connections onto the line of canonical connections. Besides, it moves metric connections onto metric connections.