# nLab pseudo-orthogonal structure

### Context

#### Riemannian geometry

Riemannian geometry

## Basic definitions

• Riemannian manifold

• moduli space of Riemannian metrics

• pseudo-Riemannian manifold

• geodesic

• Levi-Civita connection

• ## Theorems

• Poincaré conjecture-theorem
• ## Applications

• gravity

• # Contents

## Idea

The concept of pseudo-orthogonal structure is the analog of that of orthogonal structure as one generalized from Euclidean signature to Lorentzian signature.

Given a smooth manifold $X$ of dimension $n$, then a pseudo-orthogonal structure of $X$ is a reduction of the structure group of its frame bundle along the inclusion $O(n-1,1) \hookrightarrow GL(n)$ of the Lorentz group into the general linear group.

In physics, specifically in the first-order formulation of gravity, such a G-structure is often called a vielbein field.

A pseudo-orthogonal structure induces a pseudo-Riemannian metric on $X$.

Created on July 22, 2017 at 10:08:12. See the history of this page for a list of all contributions to it.