## Involve: A Journal of Mathematics

• Involve
• Volume 2, Number 5 (2009), 495-509.

### On the orbits of an orthogonal group action

#### Abstract

Let $G$ be the Lie group $SO(n,ℝ)× SO(n,ℝ)$ and let $V$ be the vector space of $n×n$ real matrices. An action of $G$ on $V$ is given by

$( g , h ) . v : = g − 1 v h , ( g , h ) ∈ G , v ∈ V .$

We consider the orbits of this group action and demonstrate a cross-section to the orbits. We then determine the stabilizer for a typical element in this cross-section and completely describe the fundamental group of an orbit of maximal dimension.

#### Article information

Source
Involve, Volume 2, Number 5 (2009), 495-509.

Dates
Accepted: 28 September 2009
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.involve/1513799207

Digital Object Identifier
doi:10.2140/involve.2009.2.495

Mathematical Reviews number (MathSciNet)
MR2601573

Zentralblatt MATH identifier
1204.22004

#### Citation

Czarnecki, Kyle; Howe, R. Michael; McTavish, Aaron. On the orbits of an orthogonal group action. Involve 2 (2009), no. 5, 495--509. doi:10.2140/involve.2009.2.495. https://projecteuclid.org/euclid.involve/1513799207

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