I've been studying differential geometry and algebraic topology for a bit, and something that keeps coming up are the manifolds $GL(n,\mathbb R),$ $SL(n,\mathbb R)$, $O(n),$ $SO(n),$ etc.

I'm wondering if there are any readily available tools to visualize these manifolds as they exist inside of $\mathbb R^{n^2},$ at least for small $n$ (maybe even just $n=2$). Of course, one can get a feeling for what these things look like by studying their topological and geometric properties (homotopy and (co)homology groups, etc.), but I'm interested specifically in how they are embedded into Euclidean space. A Google search returned no results, so I figured I would ask here.

I sort of have in mind a model of $\mathbb R^3$ and a slider so one can see $3$-D cross-sections of these manifolds, but really anything helping to visualize them would be great.

  • $\begingroup$ Vizualizing is better studied for equations, like $x^2+z^2=y^3(1-y)^3$, see here. Matrix groups are linear algebraic groups of matrices, perhaps better viewed by mathematical invariants, like dimension, fundamental group, simple, semi simple or reductive etc. $\endgroup$ – Dietrich Burde Jan 17 at 19:48
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    $\begingroup$ Well, $SL( 2)$ is a quadric hypersurface, $GL(2)$ is the complement of a quadric hypersurface, $SO(2)$ is a circle, and $O(2)$ is two disjoint copies of $SO(2)$. $\endgroup$ – Matt Samuel Jan 17 at 20:12
  • $\begingroup$ Yes, this is all true, but I'm curious about where in $\mathbb R^4$ these hypersurfaces and circles lie. Topological descriptions and invariants are exactly what I am trying to avoid when I ask this question. $\endgroup$ – D. Brogan Jan 17 at 21:07

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