Definition:Euclidean Group

Definition

Let $\struct {M, g}$ be the $n$-dimensional Euclidean space.

Let $\map E n$ be the set of all isometries of $M$.

Then $\map E n$ is called the Euclidean group (of order $n$).

Further research is required in order to fill out the details. In particular: Establish connection with action $\theta : \map O n \times \R^n \to \R^n$ and semidirect product $\R^n \rtimes_\theta \map O n$ You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by finding out more. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Research}} from the code.

Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 3$: Model Riemannian Manifolds. Euclidean Spaces