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

enter image description here

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: enter image description here

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 $x.$

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  • $\begingroup$ Hint: If a linear map $A$ fixes every basis vector in a vector space then $A=id$. $\endgroup$ – Moishe Kohan Feb 25 at 4:24
  • $\begingroup$ @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? $\endgroup$ – Hurjui Ionut Feb 25 at 11:12
  • $\begingroup$ See the 1st sentence of the proof of theorem 3.2. $\endgroup$ – Moishe Kohan Feb 25 at 15:08
  • $\begingroup$ I still don't get it, can you write down an answer with details? $\endgroup$ – Hurjui Ionut Feb 25 at 18:19
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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 $$ f_*(Y)=Y. $$

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  • $\begingroup$ 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 $G=\{1\}$? $\endgroup$ – Hurjui Ionut Feb 27 at 15:05
  • $\begingroup$ @HurjuiIonut: Do you understand the proof of Proposition 1.3? It deals with a single tensor but it also work with a finite set of tensors (such as vector fields). $\endgroup$ – Moishe Kohan Feb 27 at 17:00

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