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I cannot find some basic information on $SO(n)$ ($n$ general, not just 3) as a manifold: what is the geodesic distance between two matrices, what are the eigenfunctions and eigenvalues of the Laplace-Beltrami operator. It would be good to get a recursive definition starting from $SO(2)$, where the answers are trivial.

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The article On strain measures and the geodesic distance to $SO_n$ in the general linear group has an introductory section on the geodesic distance, which might be helpful. In general I would look into Differential Geometry, Lie Groups, and Symmetric Spaces by Sigurdur Helgason.

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  • $\begingroup$ Thank you for the references. I am interested in SO(n) just in order to use them. I am looking for a ready reference so that I can make statements like: the geodesic distance on SO(n) is given by this formula, and these are the eigenfunctions and eigenvalues for the Laplacian on SO(n), see []. Neither the article nor the book nor its sequel mention such formulas. There are formulas in a book by Vilenkin, but they are stated in terms of matrix elements of a group representation. It is a lot of effort to figure out what exactly are these and how they are connected with laplican eigenfunctions. $\endgroup$ Oct 12, 2018 at 20:24

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