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Let $E \rightarrow M$ be a vector bundle. In the book Differential Analysis on Complex Manifolds by R.O. Wells a connection on E is defined to be a map a linear map $D: \Omega(E) \rightarrow \Omega^1(E)$ such that $D(\phi s) = d\phi\otimes s + \phi Ds$.

Now the book goes to choose a local basis of sections $f = (e_1,\ldots ,e_n)$(a frame) over U and shows how one can define a connection matrix with respect to a frame. It then shows if one changes the frame with a mapping $g : U \rightarrow GL(n)$ how the connection matrix changes. For example if A is the connection matrix one finds $A(fg)=g^{−1}dg+g^{-1}A(f)g$ and defining the curvature as $F(f)=A(f) \wedge A(f)+dA(f)$ one finds $F(fg)=g^{-1}F(f)g$.

I noticed that if I considerd only transformaions $g: U \rightarrow G$ where $G$ is a matrix group and if $A(f)$ initially lies in the Lie Algebra of G then after a transformation $A(fg)=g^{−1}dg+g^{-1}A(f)g$ would also be in the Lie Algebra since the first term is the maurer cartan form and the second one is the adjoint representation.

My question is if there is some sort of name or notion for only considering connections on a vector bundle such that the connection matrix lies in the Lie-algebra of a subgroup of $GL(n)$ ? Does this have to do with structure groups ?

Qu4nt4
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1 Answers1

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Yes, there is a name for that: You can associate to a manifold a principal $Gl(n)$ bundle $P_M$ over $M$ called the bundle of frames. Let $G$ be a subgroup of $Gl(n)$ A $G$-structure defined on $M$ is a $G$-reduction of $P_M$, this is equivalent to saying that you can find transition functions of $P_M$ which take their values in $G$. A connection which takes its value in the Lie algebra of $G$ is a connection adapted to the reduction.

https://en.wikipedia.org/wiki/G-structure_on_a_manifold#Connections_on_G-structures

Tsemo Aristide
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