Checking riemann curvature tensor $C^\infty(M)$ linear in $Z$ with $R(X,Y)Z=\nabla_X\nabla_YZ-\nabla Y\nabla_XZ-\nabla_{[X,Y]}Z$

Question

Let $M$ be a smooth manifold. Given affine connection $\nabla:\Gamma(TM)\times\Gamma(TM)\to\Gamma(TM)$, I want to check that curvature tensor field $R(X,Y)Z=\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z$ is $Z$ $C^\infty(M)$ linear.

So I want to check $R(X,Y)fZ=fR(X,Y)Z$. I can surely check this in local coordinates.

Q: Is there any trick I can use to simplify my checking?

Answer 1

Use the following tips

• $[X,Y]f=X(Y(f))-Y(X(f))$
• $\nabla_X fY=X(f)Y+f\nabla_X Y$