# Question about group of automorphism of some $G$-structure.

I'm reading Kobayashi's book Transformation Groups in Differential Geometry and I don't understand a thing at page 15.

I don't understand why $$U$$ consists of transformation $$a$$ of $$M$$ that leave each $$v\in V$$ invariant and why $$(*)$$ is true? I've tried the definition of an morphism of $$G$$-strucure and in the case of $$G=\{1\}$$ we don't get any information.

I also know this proposition from the same book but I don't know if one can use it to answer my question:

Where $$K_x=(u^{-1})^*\text{K}$$ is the tensor filed on $$M$$ for any $$x\in M$$ and any $$u\in P_x.$$ i.e. $$u:\mathbb{R}^n \to T_xM$$ a frame in $$x.$$

• Hint: If a linear map $A$ fixes every basis vector in a vector space then $A=id$. – Moishe Kohan Feb 25 at 4:24
• @MoisheCohen Ok, so if I take (somehow) $K$ to be the system of linear independent vectors defining the $\{1\}$- structure I will be all set because the group from the proposition, $G=\{1\}$ but how do I do that? – Hurjui Ionut Feb 25 at 11:12
• See the 1st sentence of the proof of theorem 3.2. – Moishe Kohan Feb 25 at 15:08
• I still don't get it, can you write down an answer with details? – Hurjui Ionut Feb 25 at 18:19

This is just the matter of definitions: A 1-structure on $$M$$ (which is an $$n$$-dimensional manifold), or a parallelism on $$M$$, is a collection $$X_1,...,X_n$$ of vector fields on $$M$$ which trivialize the tangent bundle $$TM$$, i.e. such that for every $$p\in M$$, $$\{X_1(p),...,X_n(p)\}$$ is a basis of $$T_pM$$. An automorphism of this structure is a diffeomorphism $$f: M\to M$$ such that $$df_p(X_i(p))=X_i(f(p)), i=1,...,n, \forall p\in M.$$ (This is just the definition.) In other words, $$f_*(X_i)=X_i, i=1,...,n.$$ Since $$f_*: {\mathfrak X}(M)\to {\mathfrak X}(M)$$ is a linear map, for every $$Y= \sum_{i=1}^n a_i X_i, a_i\in {\mathbb R},$$ we have $$f_*(Y)=Y.$$
• I have a problem with the definition of an automorphism of $\{1\}$- structures that you gave. I know that an automorphism of a $G$-structure $P$ is a diffeo of $M$, f, such that for any $p\in M$ and any $u\in P_p$, $df_p \circ u\in P_{f(p)}$. How those two definitions are equivalent in the case $G=\{1\}$? – Hurjui Ionut Feb 27 at 15:05