# On the geometry of almost Golden Riemannian manifolds

## TL;DR

This paper explores the geometric properties of almost Golden Riemannian manifolds, focusing on special connections that reveal the structure's integrability and underlying geometry.

## Contribution

It introduces and analyzes two specific connections, the canonical and well-adapted, for almost Golden Riemannian structures, highlighting their roles in understanding integrability.

## Key findings

- Identified the canonical connection for almost Golden Riemannian manifolds.
- Defined the well-adapted connection measuring structure integrability.
- Provided insights into the geometry and structure of these manifolds.

## Abstract

An almost Golden Riemannian structure $(\varphi ,g)$ on a manifold is given by a tensor field $\varphi $ of type (1,1) satisfying the Golden section relation $\varphi ^{2}=\varphi +1$, and a pure Riemannian metric $g$, i.e., a metric satisfying $g(\varphi X,Y)=g(X,\varphi Y)$. We study connections adapted to such a structure, finding two of them, the first canonical and the well adapted, which measure the integrability of $\varphi $ and the integrability of the $G$-structure corresponding to $(\varphi ,g)$.