# Reconciling different expressions for Riemann curvature tensor

[Note: This has been crossposted to Physics SE, but I haven't found a thourough explanation there so far, so I'm posted the question here as well] I'm using Einstein's summation convention throughout. Also since I'm new to this subject, I'd be really grateful if any assertions can be shown through explicit calculations so that I can follow along and learn in the process.

I'm reading Carroll's GR notes and I'm having trouble deciphering a particular expression for the Riemann curvature tensor. The coordinate-free definition is (eq. 3.71 in Carroll's notes): $$R(X,Y)Z=\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z\tag{3.71}$$ Some more background on the first equation, based on my reading of "Semi-Riemannian Geometry: The Mathematical Language of General Relativity" by Newman:

$$R$$ is a map from $$\mathfrak{X}(M)^3$$ to $$\mathfrak{X}(M)$$ such that (3.71) holds. Newman's book doesn't really treat $$R(X,Y)$$ as a separate object (as far as I've read). $$R$$ is just treated as a map of 3 vector fields and with the weird notation $$R(X,Y)Z$$ instead of $$R(X,Y,Z)$$ (again, as far as I've read). Then there is a theorem showing that $$R$$ is multilinear in all its arguments and anti-symmetric in its first 2 arguments. And then the following assertion:

Let $$(M,\nabla)$$ be a smooth manifold with a connection, and let $$(U,(x^i))$$ be a chart on $$M$$. Then in local coordinates, $$R(\partial/\partial x^{\mu},\partial/\partial x^{\nu})\partial/\partial x^{\sigma}$$ can be expressed as: $$R\bigg(\frac{\partial}{\partial x^{\mu}},\frac{\partial}{\partial x^{\nu}}\bigg)\frac{\partial}{\partial x^{\sigma}}=R_{\ \ \sigma\mu\nu}^{\rho}\frac{\partial}{\partial x^{\rho}}$$

Based on the above, if $$V\in\mathfrak{X}(M)$$, then $$R\bigg(\frac{\partial}{\partial x^{\mu}},\frac{\partial}{\partial x^{\nu}}\bigg)V=R\bigg(\frac{\partial}{\partial x^{\mu}},\frac{\partial}{\partial x^{\nu}}\bigg)\bigg(V^{\sigma}\frac{\partial}{\partial x^{\sigma}}\bigg) \\=V^{\sigma}R\bigg(\frac{\partial}{\partial x^{\mu}},\frac{\partial}{\partial x^{\nu}}\bigg)\frac{\partial}{\partial x^{\sigma}}=V^{\sigma}R_{\ \ \sigma\mu\nu}^{\eta}\frac{\partial}{\partial x^{\eta}}$$ where the second equality holds due to multilinearity of $$R$$. Now if I act this on the $$x^{\rho}$$ coordinate function, I get $$\bigg(R\bigg(\frac{\partial}{\partial x^{\mu}},\frac{\partial}{\partial x^{\nu}}\bigg)V\bigg)(x^{\rho})\equiv\bigg(R\bigg(\frac{\partial}{\partial x^{\mu}},\frac{\partial}{\partial x^{\nu}}\bigg)V\bigg)^{\rho}=V^{\sigma}R_{\ \ \sigma\mu\nu}^{\eta}\frac{\partial}{\partial x^{\eta}}(x^{\rho}) \\=V^{\sigma}R_{\ \ \sigma\mu\nu}^{\eta}\frac{\partial x^{\rho}}{\partial x^{\eta}}=R_{\ \ \sigma\mu\nu}^{\rho}V^{\sigma}\tag{1}$$

So far, so good. Important note: I'm using the notation $$\nabla_{\mu}\equiv\nabla_{\frac{\partial}{\partial x^{\mu}}}$$, where $$\partial/\partial x^{\mu}$$ is the $$\mu$$-th basis vector field on $$(U,(x^i))$$.

Now in eq. (3.71), I can replace $$X,Y$$ by fields $$\partial_{\mu},\partial_{\nu}$$ respectively and $$Z$$ by $$V$$, then I can get the local coordinates for both sides by acting them on coordinate function $$x^{\rho}$$: $$(R(\partial_{\mu},\partial_{\nu})V)(x^{\rho})\equiv(R(\partial_{\mu},\partial_{\nu})V)^{\rho}=([\nabla_{\mu},\nabla_{\nu}]V)(x^{\rho})-(\nabla_{[\partial_{\mu},\partial_{\nu}]}V)(x^{\rho}) \\=([\nabla_{\mu},\nabla_{\nu}]V)(x^{\rho})\equiv([\nabla_{\mu},\nabla_{\nu}]V)^{\rho}\tag{2}$$

Comparing (1) and (2), I get $$R_{\ \ \sigma\mu\nu}^{\rho}V^{\sigma}=([\nabla_{\mu},\nabla_{\nu}]V)^{\rho}\tag{3}$$

The above equation completely follows from (3.71). Now in Carroll's notes, an index-based expression for the Riemann curvature tensor is also given (eq. 3.66): $$R^{\rho}_{\ \ \sigma\mu\nu}V^{\sigma}=[\nabla_{\mu},\nabla_{\nu}]V^{\rho}+T_{\mu\nu}^{\ \ \ \ \lambda}\nabla_{\lambda}V^{\rho}\tag{3.66}$$

So far torsion-free assumption hasn't been made anywhere. To reconcile (3.71) with (3.66), I have to show (comparing RHS of (3) and (3.66)): $$([\nabla_{\mu},\nabla_{\nu}]V)^{\rho}=[\nabla_{\mu},\nabla_{\nu}]V^{\rho}+T_{\mu\nu}^{\ \ \ \ \lambda}\nabla_{\lambda}V^{\rho}\tag{4}$$

This is where I'm stuck and I really cannot wrap my head around it. How do I go about proving eq. (4)? The first term on the RHS - I can only infer that it's $$[\nabla_{\mu},\nabla_{\mu}](V^{\rho})$$. If it were $$([\nabla_{\mu},\nabla_{\nu}]V)^{\rho}$$, then it would equal the LHS and the 2nd RHS term would just be zero and meaningless to include - this can't be. But then if it's $$[\nabla_{\mu},\nabla_{\mu}](V^{\rho})$$, well $$V^{\rho}$$ is a $$C^{\infty}(U)$$ function and this would evaluate to $$\partial_{\mu}\partial_{\nu}V^{\rho}-\partial_{\nu}\partial_{\mu}V^{\rho}=0$$ by equality of mixed partial derivatives.

I'm not sure where I've gone wrong in the specific approach outlined in this question. I'd appreciate any help or corrections!

EDIT: Some calculations I've done after looking at the accepted answer: $$A=\nabla_X(\nabla_YZ)=\nabla_X(\nabla_{Y^b\partial_b}Z)=\nabla_X(Y^b\nabla_{\partial_b}Z)$$ $$=X(Y^b)\nabla_{\partial_b}Z+Y^b\nabla_X(\nabla_{\partial_b}Z)=X(Y^b)\nabla_{\partial_b}Z+Y^b\nabla_{X^a\partial_a}(\nabla_{\partial_b}Z)$$ $$=X(Y^b)\nabla_{\partial_b}Z+Y^bX^a\nabla_{\partial_a}(\nabla_{\partial_b}Z)$$ $$\\$$ $$B=\nabla_{\nabla_XY}Z=\nabla_{\nabla_X(Y^b\partial_b)}Z=\nabla_{X(Y^b)\partial_b+Y^b\nabla_X\partial_b}Z$$ $$=\nabla_{X(Y^b)\partial_b}Z+\nabla_{Y^b\nabla_X\partial_b}Z=X(Y_b)\nabla_{\partial_b}Z+Y^b\nabla_{\nabla_X\partial_b}Z$$ $$=X(Y_b)\nabla_{\partial_b}Z+Y^bX^a\nabla_{\nabla_{\partial_a}\partial_b}Z$$ $$\\$$ $$A-B=X^aY^b\nabla_a\nabla_bZ \\\implies \nabla_{\partial_a}(\nabla_{\partial_b}Z)-\nabla_{\nabla_{\partial_a}\partial_b}Z=\nabla_a\nabla_bZ\tag{5}$$ So now I have to convince myself that eq. (5) is correct.

The problem in your computation is mostly notational: the usual convention distinguishes between $$\nabla_X\nabla_YZ$$ and $$X^aY^b\nabla_a\nabla_bZ$$, so that it is in fact not the standard notation to let $$\nabla_{\partial_a}\nabla_{\partial_b}Z=\nabla_a\nabla_bZ$$. Rather, the correct expression also differentiates $$\partial_b$$ along $$\partial_a$$, namely $$\nabla_{\partial_a}\nabla_{\partial_b}Z=\nabla_a\nabla_bZ+\nabla_{(\nabla_{\partial_a}\partial_b)}Z.$$ Using this formula in your calculation should give you the expression you are looking for.

To see why the above expression should be the right one, consider the following computation: $$\nabla_X(\nabla_YZ)=X^a\nabla_a(Y^b\nabla_bZ)=X^a(\nabla_aY^b)\nabla_bZ+X^aY^b\nabla_a\nabla_bZ=\nabla_{\nabla_XY}Z+X^aY^b\nabla_a\nabla_bZ$$ so that $$\nabla\nabla Z(X,Y)=\nabla\left(\nabla Z(Y)\right)(X)-\nabla Z(\nabla_XY)$$, for every vector fields $$X,Y,Z$$. The linked questions/answers provide also some more details, but it boils down to this.

Addendum: the covariant derivative $$\nabla$$ is a connection on the vector bundle $$TM\to M$$. This means that for any section $$Z$$ of $$TM$$ (i.e. a vector field) and any tangent vetor $$v$$ to $$M$$, the expression $$\nabla_vZ$$ is another section of $$TM$$. So, if $$X$$ and $$Y$$ are vector fields, it makes sense to consider $$\nabla_X\left(\nabla_Y Z\right)$$, but notice that this is not tensorial in $$X,Y$$. This would cause some problems if we defined $$\nabla_a$$ to be $$\nabla_{\partial_a}$$: not for the derivative of a single vector field, but when you iterate the covariant derivative things get more confusing, as we saw.

Anyway, a notation like $$\nabla_a\nabla_b T^c$$ usually denotes the components of the tensor $$\nabla\nabla T$$, namely we define $$\nabla_a\nabla_b T^c$$ to be the unique locally defined function such that $$\nabla\nabla T=\nabla_a\nabla_b T^c\mathrm{d}x^a\otimes\mathrm{d}x^b\otimes\partial_{x^c}$$ so $$\nabla_a\nabla_b T^c\mathrm{d}x^a\otimes\mathrm{d}x^b\otimes\partial_{x^c}=\mathrm{d}x^c\left(\nabla\nabla T(\partial_{x^a},\partial_{x^b})\right)$$.

Similarly, $$\nabla_XY=X^a\nabla_aY^b\partial_{x^b}$$, but $$\nabla_a Y^b$$ is not the derivative of the function $$Y^b$$. Rather, $$\nabla_aY^b=\mathrm{d}x^b(\nabla Y(\partial_{x^a}))$$.

• Thank you! A couple of doubts if you don't mind, I'll ask them step by step - first thing is that the equality $\nabla_X(\nabla_YZ)=X^a\nabla_a(Y^b\nabla_bZ)$ suggests to me that in general, $\nabla_XV=X^a\nabla_aV$. But I can also write $X=X^a\partial_a$, so $\nabla_XV=\nabla_{X^a\partial_a}V=X^a\nabla_{\partial_a}V$, from which I'm getting the idea that $\nabla_a$ is the same as $\nabla_{\partial_a}$. Is this fine or have I gone wrong in the above logic? And I'm guessing that even if $\nabla_a=\nabla_{\partial_a}$, I can't blindly replace $\nabla_{\partial_a}$ by $\nabla_a$ ... [cont'd] Feb 11 at 15:13
• [cont'd]... since this causes a problem when we're taking successive covariant derivatives. So then the question becomes - if I can interpret $\nabla_{\partial_a}\nabla_{\partial_b}$ as "the composition of covariant derivative w.r.t. $\partial_b$ followed by cov derivative w.r.t. $\partial_a$". Then what should I interpret $\nabla_a\nabla_b$ as? Feb 11 at 15:14
• I think the answer will be too long for a comment, I will expand my previous answer to include the details Feb 11 at 16:29
• Sure thing. In that case there's one more thing to clarify - you've equated $X^a(\nabla_aY^b)\nabla_b$ to $\nabla_{\nabla_XY}$. I was trying to figure that out - $X^a(\nabla_aY^b)\nabla_b=(X^a\nabla_aY^b)\nabla_b=(\nabla_XY^b)\nabla_b$. Whereas $\nabla_{\nabla_XY}=(\nabla_XY)^b\nabla_b$. This $(\nabla_XY^b)\nabla_b$ vs. $(\nabla_XY)^b\nabla_b$ difference is confusing - probably yet again I'm missing some notational subtlety. Feb 11 at 16:44
• I find the notation $\nabla_X\nabla_YZ$ to be confusing. So now I write either $\nabla_X(\nabla_YZ)$ or $$\nabla^2_{XY}Z = \nabla_X(\nabla_YZ) - \nabla_{\nabla_XY}Z.$$ The Riemann curvature tensor is $$R(X,Y)Z = \nabla^2_{XY}Z - \nabla^2_{YX}Z.$$ Feb 12 at 0:29