Return to Question

Attempt 3 (in response to Ben): every one of these maps $\hat{h}_{\alpha a}$ can be identified with a $G\times G'$-equivariant map $\Phi_{\alpha a}:U_{\alpha a}\times(G\times G')\to C^{\infty}(F,F')$ as follows: $$\Phi_{\alpha a}:\quad (p,g,g')\mapsto \left(f\mapsto g' \hat{h}_{\alpha a}\left(h(p),g^{-1}f\right)\right).$$ The action of $G\times G'$ on $C^{\infty}(F,F')$ is by composition $(g,g',\varphi)\mapsto g'\circ\varphi\circ g^{-1}$. The transformation rule becomes very simple: $$\boxed{\Phi_{\beta b}(p,g,g')=\Phi_{\alpha a}\left(p,t_{\beta\alpha}(p)g,t_{ba}'(h(p))g'\right).}$$ This clearly defines a global equivariant function on a principal $(G\times G')$-bundle but I want to omit that. I would like to set some restriction on these maps so that I could say that $h$ respects the $G$- and $G'$-structures in some sense. I can do it by analogy with the interpretation of a $G$-structure as a way of selecting an orbit inside the set of all isomorphisms: require all $\Phi_{\alpha a}$'s to take values in a single orbit of the action of $G\times G'$ on $C^\infty(F,F')$. This is a well-defined restriction because transition functions take values in $G\times G'$. It means that, up to a choice of local charts (compatible with the structures), the morphism "looks the same" in every fiber. Now, if $F$ and $F'$ have additional structure which selects in a natural way a subclass of maps $\mathrm{Hom}(F,F')\subset C^\infty(F,F')$ (and usually such that $G$ and $G'$ are the groups of isomorphisms preserving those structures), then we also require those orbits to be orbits of maps from $\mathrm{Hom}(F,F')$.

Attempt 3 (in response to Ben): every one of these maps $\hat{h}_{\alpha a}$ can be identified with a $G\times G'$-equivariant map $\Phi_{\alpha a}:U_{\alpha a}\times(G\times G')\to C^{\infty}(F,F')$ as follows: $$\Phi_{\alpha a}:\quad (p,g,g')\mapsto \left(f\mapsto g' \hat{h}_{\alpha a}\left(h(p),g^{-1}f\right)\right).$$ The action of $G\times G'$ on $C^{\infty}(F,F')$ is by composition $(g,g',\varphi)\mapsto g'\circ\varphi\circ g^{-1}$. The transformation rule becomes very simple: $$\boxed{\Phi_{\beta b}(p,g,g')=\Phi_{\alpha a}\left(p,t_{\beta\alpha}(p)g,t_{ba}'(h(p))g'\right).}$$ This clearly defines a global equivariant function on a principal $(G\times G')$-bundle but I want to omit that. I would like to set some restriction on these maps so that I could say that $h$ respects the $G$- and $G'$-structures in some sense. I can do it by analogy with the interpretation of a $G$-structure as a way of selecting an orbit inside the set of all isomorphisms: require all $\Phi_{\alpha a}$'s to take values in a single orbit of the action of $G\times G'$ on $C^\infty(F,F')$. This is a well-defined restriction because transition functions take values in $G\times G'$. It means that, up to a choice of local charts (compatible with the structures), the morphism "looks the same" in every fiber. Now, if $F$ and $F'$ have additional structure which selects in a natural way a subclass of maps $\mathrm{Hom}(F,F')\subset C^\infty(F,F')$ (and usually such that $G$ and $G'$ are the groups of automorphisms preserving those structures), then we also require those orbits to be orbits of maps from $\mathrm{Hom}(F,F')$.

Attempt 3 (in response to Ben): every one of these maps $\hat{h}_{\alpha a}$ can be identified with a $G\times G'$-equivariant map $\Phi_{\alpha a}:U_{\alpha a}\times(G\times G')\to C^{\infty}(F,F')$ as follows: $$\Phi_{\alpha a}:\quad (p,g,g')\mapsto \left(f\mapsto g' \hat{h}_{\alpha a}\left(h(p),g^{-1}f\right)\right).$$ The action of $G\times G'$ on $C^{\infty}(F,F')$ is by composition $(g,g',\varphi)\mapsto g'\circ\varphi\circ g^{-1}$. I would like to set some restriction on these maps so that I could say that $h$ respects the $G$- and $G'$-structures in some sense. I can do it by analogy with the interpretation of a regular $G$-structure: require all $\Phi_{\alpha a}$'s to take values in a single orbit of the action of $G\times G'$ on $C^\infty(F,F')$. This is a well-defined restriction because transition functions take values in $G\times G'$. It means that, up to a choice of local charts (compatible with the structures), the morphism "looks the same" in every fiber. Now, if $F$ and $F'$ have additional structure which selects in a natural way a subclass of maps $\mathrm{Hom}(F,F')\subset C^\infty(F,F')$ (and usually such that $G$ and $G'$ are the groups of isomorphisms preserving those structures), then we also require those orbits to be orbits of maps from $\mathrm{Hom}(F,F')$.

Attempt 3 (in response to Ben): every one of these maps $\hat{h}_{\alpha a}$ can be identified with a $G\times G'$-equivariant map $\Phi_{\alpha a}:U_{\alpha a}\times(G\times G')\to C^{\infty}(F,F')$ as follows: $$\Phi_{\alpha a}:\quad (p,g,g')\mapsto \left(f\mapsto g' \hat{h}_{\alpha a}\left(h(p),g^{-1}f\right)\right).$$ The action of $G\times G'$ on $C^{\infty}(F,F')$ is by composition $(g,g',\varphi)\mapsto g'\circ\varphi\circ g^{-1}$. The transformation rule becomes very simple: $$\boxed{\Phi_{\beta b}(p,g,g')=\Phi_{\alpha a}\left(p,t_{\beta\alpha}(p)g,t_{ba}'(h(p))g'\right).}$$ This clearly defines a global equivariant function on a principal $(G\times G')$-bundle but I want to omit that. I would like to set some restriction on these maps so that I could say that $h$ respects the $G$- and $G'$-structures in some sense. I can do it by analogy with the interpretation of a $G$-structure as a way of selecting an orbit inside the set of all isomorphisms: require all $\Phi_{\alpha a}$'s to take values in a single orbit of the action of $G\times G'$ on $C^\infty(F,F')$. This is a well-defined restriction because transition functions take values in $G\times G'$. It means that, up to a choice of local charts (compatible with the structures), the morphism "looks the same" in every fiber. Now, if $F$ and $F'$ have additional structure which selects in a natural way a subclass of maps $\mathrm{Hom}(F,F')\subset C^\infty(F,F')$ (and usually such that $G$ and $G'$ are the groups of isomorphisms preserving those structures), then we also require those orbits to be orbits of maps from $\mathrm{Hom}(F,F')$.

Attempt 3 (in response to Ben): every one of these maps $\hat{h}_{\alpha a}$ can be identified with a $G\times G'$-equivariant map $\Phi_{\alpha a}:U_{\alpha a}\times(G\times G')\to C^{\infty}(F,F')$ as follows: $$\Phi_{\alpha a}:\quad (p,g,g')\mapsto \left(f\mapsto g' \hat{h}_{\alpha a}\left(h(p),g^{-1}f\right)\right).$$ The action of $G\times G'$ on $C^{\infty}(F,F')$ is by composition $(g,g',\varphi)\mapsto g'\circ\varphi\circ g^{-1}$. I would like to set some restriction on these maps so that I could say that $h$ respects the $G$- and $G'$-structures in some sense. I can do it by analogy with the interpretation of a regular $G$-structure: require all $\Phi_{\alpha a}$'s to take values in a single orbit of the action of $G\times G'$ on $C^\infty(F,F')$. This is a well-defined restriction because transition functions take values in $G\times G'$. It means that, up to a choice of local charts (compatible with the structures), the morphism "looks the same" in every fiber.

Let's see what effect this has in the context of vector bundles. Here $G$ and $G'$ are some $\mathrm{GL}$-groups, say $G=\mathrm{GL}(V)$ and $G'=\mathrm{GL}(V')$. Then I'd like to say that a collection of maps $\varphi_p:F\to F'$ belonging to a single orbit of $\mathrm{GL}(V)\times \mathrm{GL}(V')$ acting on $C^\infty(F,F')$ is equivalent to there existing vector space structures on $F,F'$ (isomorphic to $V$ and $V'$ respectively) which make all of these maps linear. This then leads to the standard definition of vector bundles. The only remaining issue seems to be that linearity still looks like an external restriction. The only fix I see is to require $\Phi_{\alpha a}$ to take values not just in any orbit, but in an orbit of a linear map $F\to F'$. But then how does this generalize to other $G$-structures?

Attempt 3 (in response to Ben): every one of these maps $\hat{h}_{\alpha a}$ can be identified with a $G\times G'$-equivariant map $\Phi_{\alpha a}:U_{\alpha a}\times(G\times G')\to C^{\infty}(F,F')$ as follows: $$\Phi_{\alpha a}:\quad (p,g,g')\mapsto \left(f\mapsto g' \hat{h}_{\alpha a}\left(h(p),g^{-1}f\right)\right).$$ The action of $G\times G'$ on $C^{\infty}(F,F')$ is by composition $(g,g',\varphi)\mapsto g'\circ\varphi\circ g^{-1}$. I would like to set some restriction on these maps so that I could say that $h$ respects the $G$- and $G'$-structures in some sense. I can do it by analogy with the interpretation of a regular $G$-structure: require all $\Phi_{\alpha a}$'s to take values in a single orbit of the action of $G\times G'$ on $C^\infty(F,F')$. This is a well-defined restriction because transition functions take values in $G\times G'$. It means that, up to a choice of local charts (compatible with the structures), the morphism "looks the same" in every fiber. Now, if $F$ and $F'$ have additional structure which selects in a natural way a subclass of maps $\mathrm{Hom}(F,F')\subset C^\infty(F,F')$ (and usually such that $G$ and $G'$ are the groups of isomorphisms preserving those structures), then we also require those orbits to be orbits of maps from $\mathrm{Hom}(F,F')$.

Let's see what effect this has in the context of vector bundles. Here $G$ and $G'$ are some $\mathrm{GL}$-groups, say $G=\mathrm{GL}(F)$ and $G'=\mathrm{GL}(F')$. Then I'd like to say that a collection of maps $\varphi_p:F\to F'$ belonging to an orbit of a linear map under the action of $\mathrm{GL}(F)\times \mathrm{GL}(F')$ on $C^\infty(F,F')$ is equivalent to these maps being linear. This then leads to the standard definition of vector bundles. So the solution is that the additional structure of a vector space on the model fiber canonically selects of class of fiberwise maps, and these generate bundle morphisms.