# Flat metric

*This article defines a property that makes sense for a Riemannian metric over a differential manifold*

*This is the property of the following curvature being everywhere zero*: sectional curvature

## Contents

## Definition

### Symbol-free definition

A Riemannian metric on a differential manifold is said to be a **flat metric** if it satisfies the following equivalent conditions:

- The sectional curvature is identically zero, viz the sectional curvature is zero for every tangent plane at every point.
- The Riemann curvature tensor of the Levi-Civita connection is the zero map, viz vanishes everywhere

## Relation with other properties

### Weaker properties

- Constant-curvature metric
- Ricci-flat metric
- Einstein metric
- Constant-scalar curvature metric
- Conformally flat metric

## Facts

### Holonomy

The restricted holonomy group of a flat metric is trivial. In other words, any loop homotopic to the constant loop, has trivial holonomy. Thus, the holonomy group is a homomorphic image of the fundamental group, and gives rise to a linear representation of the fundamental group.

### Minding's theorem

`Further information: Minding's theorem`

This result says that two smooth surfaces with the same constant curvature are locally isometric.

## Metaproperties

### Direct product-closedness

*This property of a Riemannian metric on a differential manifold is closed under taking direct products of manifolds*

The direct product of two Riemannian manifolds, both equipped with a flat metric, also gets the flat metric. Thus, since every curve is a flat manifold, any manifold obtained as a direct product of curves is also flat.