Is the set of average-position preserving transformations a Lie group - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T16:52:43Z http://mathoverflow.net/feeds/question/71836 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/71836/is-the-set-of-average-position-preserving-transformations-a-lie-group Is the set of average-position preserving transformations a Lie group hoj201 2011-08-01T21:53:37Z 2011-08-01T21:53:37Z <p>Let $M$ be a compact subset in $\mathbb{R}^n$ and $\mu$ a volume form on $M$. Let $x_i$ denote the function corresponding to the $i$-coordinate. Does the set of diffeomorphisms satisfying $$\int_M{x_i \mu } = \int_M{ \varphi_*(x_i) \mu }, \quad i = 1,\dots,n$$ form a Lie group? In other words, is the set of transformations that preserve the $\mu$-average position a Lie group? Certainly the isotropy group of the $\mu$-average position is a Lie-Group, however this set seems to incude a little more than the isotropy group.</p>