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In Milnor's Morse Theory, he defines the curvature $R$ of an affine connection $\vdash$ as $$R(X,Y)Z=-X\vdash(Y\vdash Z)+Y\vdash(X\vdash Z)+[X,Y]\vdash Z.$$ In his proof of Lemma 9.3 part (2), which states that $R(X,Y)Z+R(Y,Z)X+R(Z,X)Y=0$, he writes that, "Since all three terms of (2) are tensors, it is sufficient to prove (2) when the bracket products $[X,Y]$, $[X,Z]$, and $[Y,Z]$ are all zero."

Now, I understand how to prove this lemma without using this simplification, so I don't really need a proof of this lemma. But I don't really understand how this "wlog" follows at all from tensor-ness.

I did see the following: Bianchi identity proof : why we can consider $[X,Y]=[Y,Z]=[X,Z]=0$? However, as I commented on that post a couple days ago, it seems that the answer implies that all vectors fields $X,Y$ have $[X,Y]=0$, which is obviously false.

Thanks!

boink
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    What makes you think that $\left[\frac{\partial}{\partial x^i},\frac{\partial}{\partial x^j}\right]=0$ implies that any two vector fields commute? – Kajelad Jan 30 '21 at 07:53

1 Answers1

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The lemma i know is:

For $v_1,\dots,v_n\in T_pM$ there are vectorfields $ X_1,\dots, X_n\in\Gamma TM$ with $v_i= (X_i)_p$ and $[ X_i, X_j]_q=0$ for all $q$ in a neighbourhood of $p$.

This can be achieved by writing the $v_i$ as linear combinations of some coordinate vectorfields and then multiplying the linear combinations of coordinate vectorfields by a bump function which is $1$ in a neighborhood of $p$.

Now tensoriality of a map $\beta:(\Gamma TM)^n\to\Gamma E$ implies that $\beta(X_1,\dots, X_n)_p$ only depends on the values of the $X_i$ at $p$, so to show for example $\beta(X_1,\dots, X_n)_p=0$, by the above lemma one can assume $[X_i,X_j]=0$ in a neighboorhod of $p$.

This of course is weaker than assuming $[X_i,X_j]=0$ everywhere, but in most cases sufficient, since if $\nabla$ is a connection then $(\nabla_XY)_p$ only depends on the value of $X$ at $p$ and the values of $Y$ in a neighborhood of $p$.

Claire
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