Curvature is tensorial

This article gives the statement, and possibly proof, that a map constructed in a certain way is tensorial
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Statement

Let $\nabla$ be a connection on a vector bundle $E$ over a differential manifold $M$ . The Riemann curvature tensor of $\nabla$ is given as a map $\Gamma(TM) \otimes \Gamma(TM) \otimes \Gamma(E) \to \Gamma(E)$ defined by:

$R(X,Y)Z = \nabla_X\nabla_Y Z - \nabla_Y \nabla_X Z - \nabla_{[X,Y]} Z$

We claim that $R$ is a tensorial map in each of the variables $X,Y,Z$ .

Related facts

Facts used

Fact no.	Name	Statement with symbols
1	Any connection is $C^\infty$ -linear in its subscript argument	$\nabla_{fA} = f\nabla_A$ for any $C^\infty$ -function $f$ and vector field $A$ .
2	The Leibniz-like axiom that is part of the definition of a connection	For a function $f$ and vector fields $A,B$ , and a connection $\nabla$ , we have $\nabla_A(fB) = (Af)(B) + f\nabla_A(B)$
3	Corollary of Leibniz rule for Lie bracket (in turn follows from Leibniz rule for derivations	For a function $f$ and vector fields $X,Y$ : $\! f[X,Y] = [fX,Y] + (Yf)X$ $\! f[X,Y] = [X,fY] - (Xf)Y$

Proof

To prove tensoriality in a variable, it suffices to show $C^\infty$ -linearity in that variable. This is because linearity in $C^\infty$ -functions guarantees linearity in a function that is 1 at exactly one point, and zero at others.

The proofs for $X$ and $Y$ are analogous, and rely on manipulation of the Lie bracket $[fX,Y]$ and the property of a connection being $C^\infty$ in the subscript vector. These proofs do not involve any explicit use of $Z$ . The proof for $Z$ relies simply on repeated application of the product rule, and the fact that $XY - YX = [X,Y]$ .

Tensoriality in the first variable

Given: $f:M \to \R$ is a $C^\infty$ -function.

To prove: $\! R(fX,Y) = f R(X,Y)$ , or more explicitly, $\! \nabla_{fX}\nabla_Y - \nabla_Y \nabla_{fX} - \nabla_{[fX,Y]} = f(\nabla_X\nabla_Y - \nabla_Y\nabla_X - \nabla_{[X,Y]}$

We start out with the left side:

$\nabla_{fX}\nabla_Y - \nabla_Y \nabla_{fX} - \nabla_{[fX,Y]}$

Each step below is obtained from the previous one via some manipulation explained along side.

Step no.	Current status of left side	Facts/properties used	Specific rewrites
1	$f\nabla_X\nabla_Y - \nabla_Y (f \nabla_X) - \nabla_{[fX,Y]}$	Fact (1): $\nabla$ is $C^\infty$ -linear in its subscript argument.	$\nabla_{fX} \to f\nabla_X$
2	$f\nabla_X\nabla_Y - (Yf)\nabla_X - f \nabla_Y\nabla_X - \nabla_{[fX,Y]}$	Fact (2)	$\nabla_Y(f \nabla_X) \to (Yf)\nabla_X + f\nabla_Y\nabla_X$ . To understand this more clearly imagine an input $Z$ to the whole expression, so that the rewrite becomes $\nabla_Y(f \nabla_X(Z)) \to (Yf)\nabla_X(Z) + f\nabla_Y\nabla_X(Z)$ . In the notation of fact (3), $A = Y$ , $f = f$ , and $B = \nabla_X(Z)$ .
3	$f(\nabla_X\nabla_Y - \nabla_Y\nabla_X) - \nabla_{(Yf)X} - \nabla_{[fX,Y]}$	Fact (1)	$(Yf)\nabla_X \to \nabla_{(Yf)X}$
4	$f(\nabla_X\nabla_Y - \nabla_Y\nabla_X) - \nabla_{(Yf)X + [fX,Y]}$	$\nabla$ is additive in its subscript argument	$\nabla_{(Yf)X} + \nabla_{[fX,Y]} = \nabla_{(Yf)X + [fX,Y]}$
5	$f(\nabla_X\nabla_Y - \nabla_Y\nabla_X) - \nabla_{f[X,Y]}$	Fact (3)	$[fX,Y] + (Yf)X \to f[X,Y]$
6	$f(\nabla_X\nabla_Y - \nabla_Y\nabla_X - \nabla_{[X,Y]})$	Fact (1)	$\nabla_{f[X,Y]} \to f\nabla_{[X,Y]}$

Tensoriality in the second variable

Given: $f:M \to \R$ is a $C^\infty$ -function.

To prove: $\! R(X,fY) = f R(X,Y)$ , or more explicitly, $\nabla_X\nabla_{fY} - \nabla_{fY}\nabla_X - \nabla_{[X,fY]} = f (\nabla_X\nabla_Y - \nabla_Y\nabla_X - \nabla_{[X,Y]}$ .

We start out with the left side:

$\nabla_X\nabla_{fY} - \nabla_{fY}\nabla_X - \nabla_{[X,fY]}$

Each step below is obtained from the previous one via some manipulation explained along side.

Step no.	Current status of left side	Facts/properties used	Specific rewrites
1	$\nabla_X(f\nabla_Y) - f\nabla_Y\nabla_X - \nabla_{[X,fY]}$	Fact (1)	$\nabla_{fY} \to f\nabla_Y$ .
2	$(Xf)\nabla_Y + f(\nabla_X\nabla_Y) - f\nabla_Y\nabla_X - \nabla_{[X,fY]}$	Fact (2)	$\nabla_X(f\nabla_Y) \to (Xf)\nabla_Y + f(\nabla_X\nabla_Y)$ . To make this more concrete, imagine an input $Z$ . Then, the rewrite becomes $\nabla_X(f\nabla_Y(Z)) \to (Xf)\nabla_Y(X) + f(\nabla_X\nabla_Y(Z))$ . This comes setting $A = X$ , $f = f$ , $B = \nabla_YZ$ in Fact (3).
3	$f(\nabla_X\nabla_Y - \nabla_Y\nabla_X) - \nabla_{[X,fY]} + \nabla_{(Xf)Y}$	Fact (1)	$(Xf)\nabla_Y \to \nabla_{(Xf)Y}$
4	$f(\nabla_X\nabla_Y - \nabla_Y\nabla_X) - \nabla_{[X,fY] - (Xf)Y}$	$\nabla$ is additive in its subscript argument.	$\nabla_{[X,fY]} - \nabla_{(Xf)Y} \to \nabla_{[X,fY] - (Xf)Y}$ .
5	$f(\nabla_X\nabla_Y - \nabla_Y\nabla_X) - \nabla_{f[X,Y]}$	Fact (3)	$[X,fY] - (Xf)Y \to f[X,Y]$
6	$f(\nabla_X\nabla_Y - \nabla_Y\nabla_X - \nabla_{[X,Y]})$	Fact (1)	$\nabla_{f[X,Y]} \to f\nabla_{[X,Y]}$

Tensoriality in the third variable

Given: A $C^\infty$ -function $f:M \to \R$ .

To prove: $\! R(X,Y) (fZ) = f R(X,Y) Z$ . More explicitly, $\! \nabla_X\nabla_Y(fZ) - \nabla_Y\nabla_X(fZ) - \nabla_{[X,Y]}(fZ) = f (\nabla_X\nabla_Y - \nabla_Y\nabla_X - \nabla_{[X,Y]})Z + ((XY - YX - [X,Y])f)Z$ .

We start out with the left side:

$\nabla_X\nabla_Y(fZ) - \nabla_Y\nabla_X(fZ) - \nabla_{[X,Y]}(fZ)$

Each step below is obtained from the previous one via some manipulation explained along side.

Step no.	Current status of left side	Facts/properties used	Specific rewrites
1	$\! \nabla_X( (Yf)(Z) + f \nabla_YZ) - \nabla_Y ((Xf)Z + f \nabla_XZ) - f \nabla_{[X,Y]}Z - ([X,Y]f) Z$	Fact (2)	$\nabla_Y(fZ) \to (Yf)(Z) + f\nabla_YZ$ and $\nabla_X(fZ) \to (Xf)Z + f\nabla_XZ$
2	$\! (XYf)(Z) + (Yf) \nabla_XZ + (Xf) \nabla_YZ + f \nabla_X\nabla_YZ - (YXf)Z - (Xf) \nabla_YZ - (Yf) \nabla_XZ -f \nabla_Y\nabla_XZ - f \nabla_{[X,Y]}Z - ([X,Y]f) Z$	Fact (2)	$\nabla_X((Yf)Z) \to X((Yf)Z) + (Yf)\nabla_XZ$ , etc.
3	$f (\nabla_X\nabla_Y - \nabla_Y\nabla_X - \nabla_{[X,Y]})Z + ((XY - YX - [X,Y])f)Z$	--	cancellations
4	$f (\nabla_X\nabla_Y - \nabla_Y\nabla_X - \nabla_{[X,Y]})Z + ((XY - YX - [X,Y])f)Z$	use $[X,Y] = XY - YX$ , definition	cancellation