# Curvature

The exterior covariant derivative $${\mathrm{D}}$$ parallel transports its values on the boundary before summing them, and therefore we do not expect it to mimic the property $${\mathrm{d}^{2}=0}$$. Indeed it does not; instead, for a vector field $${w}$$ viewed as a vector-valued 0-form $${\vec{w}}$$, we have

$$\displaystyle \left(\mathrm{D}^{2}\vec{w}\right)(u,v)\equiv\check{R}\left(u,v\right)\vec{w}=\nabla_{u}\nabla_{v}w-\nabla_{v}\nabla_{u}w-\nabla_{\left[u,v\right]}w,$$

which defines the curvature 2-form $${\check{R}}$$, which is $${gl(\mathbb{R}^{n})}$$-valued. From its definition, $${\check{R}\vec{w}}$$ is a frame-independent quantity, and thus if $${\vec{w}}$$ is considered as a vector-valued 0-form, $${\check{R}}$$ is frame-independent as well. In the (more common) case that we view $${\vec{w}}$$ as a frame-dependent $${\mathbb{R}^{n}}$$-valued 0-form, $${\check{R}}$$ must be considered to be $${gl(n,\mathbb{R})}$$-valued, and is thus a frame-dependent matrix, transforming under a (frame-dependent) $${GL(n,\mathbb{R})}$$-valued 0-form $${\check{\gamma}^{-1}}$$ change of frame like

$$\displaystyle \check{R}^{\prime}=\check{\gamma}\check{R}\check{\gamma}^{-1}.$$

A connection with zero curvature is called flat, as is any region of $${M}$$ with a flat connection.

For a general $${\mathbb{R}^{n}}$$-valued form $${\vec{\varphi}}$$ it is not hard to arrive at an expression for $${\check{R}}$$ in terms of the connection:

$$\displaystyle \mathrm{D}^{2}\vec{\varphi}=\left(\mathrm{d}\check{\Gamma}+\check{\Gamma}\wedge\check{\Gamma}\right)\wedge\vec{\varphi}=\check{R}\wedge\vec{\varphi}$$

Note that $${\mathrm{D}\check{\Gamma}=\mathrm{d}\check{\Gamma}+\check{\Gamma}[\wedge]\check{\Gamma}}$$ is a similar but distinct construction, since $${(\check{\Gamma}\wedge\check{\Gamma})\left(v,w\right)=\check{\Gamma}\left(v\right)\check{\Gamma}\left(w\right)-\check{\Gamma}\left(w\right)\check{\Gamma}\left(v\right)}$$, while $${(\check{\Gamma}[\wedge]\check{\Gamma})\left(v,w\right)=[\check{\Gamma}\left(v\right),\check{\Gamma}\left(w\right)]-[\check{\Gamma}\left(w\right),\check{\Gamma}\left(v\right)]=2(\check{\Gamma}\wedge\check{\Gamma})\left(v,w\right)}$$. Thus we can equivalently define

\begin{aligned}\check{R} & \equiv\mathrm{d}\check{\Gamma}+\check{\Gamma}\wedge\check{\Gamma}\\ & =\mathrm{d}\check{\Gamma}+\frac{1}{2}\check{\Gamma}[\wedge]\check{\Gamma}. \end{aligned}

The definition of $${\check{R}}$$ in terms of $${\check{\Gamma}}$$ is sometimes called Cartan’s second structure equation. An immediate property from the definition of $${\check{R}}$$ is $${\check{R}(u,v)=-\check{R}(v,u)}$$, which allows us to write e.g. for a vector-valued 1-form $${\vec{\varphi}}$$

\begin{aligned}\left(\mathrm{D}^{2}\vec{\varphi}\right)(u,v,w) & =\left(\check{R}\wedge\vec{\varphi}\right)\left(u,v,w\right)\\ & =\check{R}\left(u,v\right)\vec{\varphi}(w)+\check{R}\left(v,w\right)\vec{\varphi}(u)+\check{R}\left(w,u\right)\vec{\varphi}(v). \end{aligned}

Constructing the same picture as we did for the double exterior derivative, we put $${\mathrm{D}^{2}\vec{w}\equiv\mathrm{D}\vec{\varphi}}$$, where $${\vec{\varphi}(v)\equiv\mathrm{D}\vec{w}(v)=\nabla_{v}w}$$. Expanding both derivatives in terms of parallel transport, we find in the following figure that as we sum values around the boundary of the surface defined by its arguments, $${\mathrm{D}^{2}}$$ fails to cancel the endpoint and starting point at the far corner. Examining the values of these non-canceling points, we can view the curvature as “the difference between $${w}$$ when parallel transported around the two opposite edges of the boundary of the surface defined by its arguments.” The above depicts how $${\check{R}\left(u,v\right)\vec{w}=\left(\mathrm{D}^{2}\vec{w}\right)(u,v)}$$ is “the difference between $${w}$$ when parallel transported around the two opposite edges of the boundary of the surface defined by its arguments.” In the figure we assume vanishing Lie bracket for simplicity, so that $${v\left|_{p+\varepsilon u+\varepsilon v}\right.=v\left|_{p+\varepsilon v+\varepsilon u}\right.}$$.

In terms of the connection, we can use the path-ordered exponential formulation to examine the parallel transporter around the closed path $${L\equiv\partial S}$$ defined by the surface $${S\equiv\left(\varepsilon u\wedge\varepsilon v\right)}$$ to order $${\varepsilon^{2}}$$. This calculation after some work (see  pp. 51-53) yields

$$\displaystyle \parallel_{L}(w)=P\textrm{exp}\left(-\int_{L}\check{\Gamma}\right)\vec{w}=w-\int_{S}\left(\mathrm{d}\check{\Gamma}+\check{\Gamma}\wedge\check{\Gamma}\right)\vec{w}=w-\varepsilon^{2}\check{R}\left(u,v\right)\vec{w},$$

where we have dropped the indices since $${L}$$ is a closed path and thus $${\parallel_{L}}$$ is basis-independent. Thus the curvature can be viewed as “the difference between $${w}$$ and its parallel transport around the boundary of the surface defined by its arguments.” The above depicts how $${\check{R}\left(u,v\right)\vec{w}}$$ is “the difference between $${w}$$ and its parallel transport around the boundary of the surface defined by its arguments.”

As these pictures suggest, one can verify algebraically that the value of $${\check{R}\left(u,v\right)\vec{w}}$$ at a point $${p}$$ only depends upon the value of $${w}$$ at $${p}$$, even though it can be defined in terms of $${\nabla w}$$, which depends upon nearby values of $${w}$$. Similarly, $${\check{R}\left(u,v\right)\vec{w}}$$ at a point $${p}$$ only depends upon the values of $${u}$$ and $${v}$$ at $${p}$$, even though it can be defined in terms of $${[u,v]}$$, which depends upon their vector field values (note that $${\nabla_{u}\nabla_{v}w}$$ depends upon the vector field values of both $${v}$$ and $${w}$$). Finally, $${\check{R}}$$ (as a $${gl(\mathbb{R}^{n})}$$-valued 2-form) is frame-independent, even though it can be defined in terms of $${\check{\Gamma}}$$, which is not. Thus the curvature can be viewed as a tensor of type $${\left(1,3\right)}$$, called the Riemann curvature tensor (AKA Riemann tensor, curvature tensor, Riemann–Christoffel tensor):

\begin{aligned}R^{c}{}_{dab}u^{a}v^{b}w^{d} & = u^{a}\nabla_{a}\left(v^{b}\nabla_{b}w^{c}\right)-v^{b}\nabla_{b}\left(u^{a}\nabla_{a}w^{c}\right)-[u,v]^{d}\nabla_{d}w^{c}\\ & =u^{a}v^{b}\nabla_{a}\nabla_{b}w^{c}-u^{a}v^{b}\nabla_{b}\nabla_{a}w^{c}+T^{d}{}_{ab}u^{a}v^{b}\nabla_{d}w^{c}\\ \Rightarrow R^{c}{}_{dab}w^{d} & =\left(\nabla_{a}\nabla_{b}-\nabla_{b}\nabla_{a}+T^{d}{}_{ab}\nabla_{d}\right)w^{c} \end{aligned}

Here we have used the Leibniz rule and recalled that $${[u,v]^{d}=u^{a}\nabla_{a}v^{d}-v^{b}\nabla_{b}u^{d}-T^{d}{}_{ab}u^{a}v^{b}}$$. To obtain an expression in terms of the connection coefficients, we first examine the double covariant derivative, recalling that $${\nabla_{b}w^{c}}$$ is a tensor:

\begin{aligned}\nabla_{a}\left(\nabla_{b}w^{c}\right) & =\partial_{a}\nabla_{b}w^{c}+\Gamma^{c}{}_{fa}\nabla_{b}w^{f}-\Gamma^{f}{}_{ba}\nabla_{f}w^{c}\\ & =\partial_{a}\partial_{b}w^{c}+\partial_{a}(\Gamma^{c}{}_{fb}w^{f})\\ & \phantom{{}=}+\Gamma^{c}{}_{fa}\partial_{b}w^{f}+\Gamma^{c}{}_{fa}\Gamma^{f}{}_{gb}w^{g}-\Gamma^{f}{}_{ba}\nabla_{f}w^{c}\\ & =\partial_{a}\partial_{b}w^{c}+\partial_{a}\Gamma^{c}{}_{fb}w^{f}\\ & \phantom{{}=}+\Gamma^{c}{}_{fb}\partial_{a}w^{f}+\Gamma^{c}{}_{fa}\partial_{b}w^{f}\\ & \phantom{{}=}+\Gamma^{c}{}_{fa}\Gamma^{f}{}_{gb}w^{g}-\Gamma^{f}{}_{ba}\nabla_{f}w^{c}. \end{aligned}

When we subtract the same expression with $${a}$$ and $${b}$$ reversed, we recognize that for the functions $${w^{c}}$$ we have $${\partial_{a}\partial_{b}w^{c}-\partial_{b}\partial_{a}w^{c}=[e_{a},e_{b}]^{d}\partial_{d}w^{c}}$$, that the second line $${\Gamma^{c}{}_{fb}\partial_{a}w^{f}+\Gamma^{c}{}_{fa}\partial_{b}w^{f}}$$ vanishes, and that $${\Gamma^{f}{}_{ba}-\Gamma^{f}{}_{ab}=[e_{a},e_{b}]^{f}+T^{f}{}_{ab}}$$, so that

\begin{aligned}\left(\nabla_{a}\nabla_{b}-\nabla_{b}\nabla_{a}\right)w^{c} & =[e_{a},e_{b}]^{d}\partial_{d}w^{c}+\partial_{a}\Gamma^{c}{}_{fb}w^{f}-\partial_{b}\Gamma^{c}{}_{fa}w^{f}\\ & \phantom{{}=}+\Gamma^{c}{}_{fa}\Gamma^{f}{}_{gb}w^{g}-\Gamma^{c}{}_{fb}\Gamma^{f}{}_{ga}w^{g}\\ & \phantom{{}=}-\left([e_{a},e_{b}]^{f}+T^{f}{}_{ab}\right)\nabla_{f}w^{c}, \end{aligned}

and thus relabeling dummy indices to obtain an expression in terms of $${w^{d}}$$, we arrive at

\begin{aligned}R^{c}{}_{dab}w^{d} & =\left(\nabla_{a}\nabla_{b}-\nabla_{b}\nabla_{a}+T^{d}{}_{ab}\nabla_{d}\right)w^{c}\\ & =\left(\partial_{a}\Gamma^{c}{}_{db}-\partial_{b}\Gamma^{c}{}_{da}+\Gamma^{c}{}_{fa}\Gamma^{f}{}_{db}-\Gamma^{c}{}_{fb}\Gamma^{f}{}_{da}-[e_{a},e_{b}]^{f}\Gamma^{c}{}_{df}\right)w^{d}. \end{aligned}

This expression follows much more directly from the expression $${\check{R}\equiv\mathrm{d}\check{\Gamma}+\check{\Gamma}\wedge\check{\Gamma}}$$, but the above derivation from the covariant derivative expression is included here to clarify other presentations which are sometimes obscured by the quirks of index notation for covariant derivatives.

 Δ The derivation above makes clear how the expression for the curvature in terms of the covariant derivative simplifies to $${R^{c}{}_{dab}w^{d}=\left(\nabla_{a}\nabla_{b}-\nabla_{b}\nabla_{a}\right)w^{c}}$$ for zero torsion but is unchanged in a holonomic frame, while in contrast the expression in terms of the connection coefficients is unchanged for zero torsion but in a holonomic frame simplifies to omit the term $${[e_{a},e_{b}]^{f}\Gamma^{c}{}_{df}w^{d}}$$.
 Δ Note that the sign and the order of indices of $${R}$$ as a tensor are not at all consistent across the literature.