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2 corrected formula

The examples you give of Lie groups associated to geometric objects are the structure groups of the tangent bundles of manifolds. The tangent bundle is locally trivial, so you have a covering of your manifold and isomorphisms *over the $U_i$* (i.e. commuting with the projections to the $U_i$) $\varphi_i:TM|_U \rightarrow \mathbb{R}^n \times U_i$. On the intersections this gives you isomorphisms *over $U_i \cap U_j$* $\varphi_j^{-1} \varphi_i \circ \varphi_i: varphi_j^{-1}: \mathbb{R}^n \times U_i \cap U_j \rightarrow leftarrow TM|_{U_i \mathbb{R}^n cap U_j} \rightarrow\mathbb{R}^n \times U_i \cap U_j$, i.e. isomorphisms $g_{ij}: \mathbb{R}^n \rightarrow \mathbb{R}^n$. These can be seen as gluing data of your bundle. The "structure group" of your bundle tells you what gluing data you allow: You could require the $g_{ij}$ to be in $GL(n), O(n), SO(n), U(n), Sp(n)$ and so on. The smaller this group of gluing data is, the more special is your bundle, e.g. if you can glue your tangent bundle only using maps from $SO(n)$, then your manifold will be orientable. The buzz word to google for is "reduction of structure group".

I am not sure what you have heard about pseudogroups but there is one consisting of the $\varphi_i$ - open patches homeomorphic to $\mathbb{R}^n$ are such that the tangent bundle is trivial over them, so if you know how to glue your manifold from such patches, you get the relation to the Lie group as above.

Of course you can embed every Lie group into some $GL_n$ and require the $g_{ij}$ to lie in that subgroup, but that seems somewhat arbitrary.

Finally some references on the Erlangen program are Sharpe's book and this, where you can look for more ingredients of the big picture.

1

The examples you give of Lie groups associated to geometric objects are the structure groups of the tangent bundles of manifolds. The tangent bundle is locally trivial, so you have a covering of your manifold and isomorphisms *over the $U_i$* (i.e. commuting with the projections to the $U_i$) $\varphi_i:TM|_U \rightarrow \mathbb{R}^n \times U_i$. On the intersections this gives you isomorphisms *over $U_i \cap U_j$* $\varphi_j^{-1} \circ \varphi_i: \mathbb{R}^n \times U_i \cap U_j \rightarrow \mathbb{R}^n \times U_i \cap U_j$, i.e. isomorphisms $g_{ij}: \mathbb{R}^n \rightarrow \mathbb{R}^n$. These can be seen as gluing data of your bundle. The "structure group" of your bundle tells you what gluing data you allow: You could require the $g_{ij}$ to be in $GL(n), O(n), SO(n), U(n), Sp(n)$ and so on. The smaller this group of gluing data is, the more special is your bundle, e.g. if you can glue your tangent bundle only using maps from $SO(n)$, then your manifold will be orientable. The buzz word to google for is "reduction of structure group".

I am not sure what you have heard about pseudogroups but there is one consisting of the $\varphi_i$ - open patches homeomorphic to $\mathbb{R}^n$ are such that the tangent bundle is trivial over them, so if you know how to glue your manifold from such patches, you get the relation to the Lie group as above.

Of course you can embed every Lie group into some $GL_n$ and require the $g_{ij}$ to lie in that subgroup, but that seems somewhat arbitrary.

Finally some references on the Erlangen program are Sharpe's book and this, where you can look for more ingredients of the big picture.