Corollary of Leibniz rule for Lie bracket

Statement

This is an identity that uses the Leibniz rule to measure the failure of the Lie bracket operation from being $C^{\infty }$-linear.

Let $X,Y$ be smooth vector fields on a differential manifold $M$ and $f$ be in $C^{\infty }(M)$. We then have:

$\!f[X,Y]=[fX,Y]+(Yf)X$

$\!f[X,Y]=[X,fY]-(Xf)Y$

Applications

• Curvature is tensorial
• Torsion is tensorial

Proof

First identity

We prove this by showing that for any $g\in C^{\infty }(M)$, both sides evaluate to the same thing. Let's do this. Simplifying the right side yields:

$f(X(Yg)-Y(Xg))-f(X(Yg))+Y((fX)g)=Y(f(Xg))-f(Y(Xg))$

Applying the Leibniz rule for $Y$ on the product of functions $f$ and $Xg$, this simplifies to:

$(Yf)(Xg)+Y(Xg)(f)-f(Y(Xg))=(Yf)(Xg)$

which is precisely equal to the left side.

Second identity

We prove this as well by taking any test function $g\in C^{\infty }(M)$. Simplifying the right side yields:

$[X,fY]g-(Xf)(Yg)=X(f(Yg))-f(YXg)-(Xf)(Yg)=(Xf)(Yg)+f(XYg)-f(YXg)-(Xf)(Yg)=f[X,Y]g$

which is the same as the left side.