Notes on the pullback connection

In an effort to keep myself organized, I decided I should type up some notes I have laying around. These calculations are not enough to be a journal article (or even a note on the arXiv), but slightly non-trivial, so I don’t want to lose them.


This note is on a construction called the pullback connection. Consider a manifold and let there be a one-parameter family of diffeomorphisms The one-parameter family is of course a Lie group under composition, and The diffeomorphism induces a number of maps, e.g. the pullback map and the pushforward map sometimes written While the pullback and pushforward are defined for general maps between manifolds, since this one is a diffeomorphism from the manifold to itself, the pullback and pushforward are inverses,

Let be the vector field tangent to the one-dimensional orbits of Recall that the Lie derivative is defined as \begin{align} \label{eq:lie-deriv-def} \lie_{v} T = \frac{d}{dt} \Big[ \varphi_{t}^{*} T \Big]_{t=0} \,. \end{align} Conversely, we can think of the pullback as exponentiation of the Lie derivative, .

Now let there be a connection on this manifold, so that for any vector field is a map from the space of tensor fields of rank to the same space; that satisfies the Leibniz property, and so on; and in particular that it only involves but not derivatives of (one example of such a connection is of course the Levi-Civita connection of a metric, but we will get to this later).

Given the connection and a diffeomorphism there is a new connection we can construct: the pullback connection, We define it as follows: \begin{align} \label{eq:pullback-connection-defn} (\varphi_{t}^{*}\cd)_{X} T \equiv \varphi_{t}^{*} \left[ \cd_{d\varphi_{t}(X)} \left( \varphi_{t*} T \right) \right] \,. \end{align} In words, this definition is: take tensor and push it forward along the map; take its derivative with your given connection , but in the direction given by the pushed-forward version of ; then take the whole result and pull it back.

As we know, the space of connections is an affine space, and there are connection coefficients which are the difference between two particular connections. A natural question would be: given and what is the difference

For a Killing vector field

I will address the general case below, but first specialize to the case when is the Levi-Civita connection of some metric , and is an isometry of , i.e. that \begin{align} \label{eq:g-isometry} g = \varphi_{t} g \,. \end{align} In this case, we see that \begin{align} \label{eq:pullback-conn-of-g-vanishes} (\varphi_{t}^{*}\cd)_{X} g = 0 \quad \forall X\,. \end{align} This is a direct application of the definition of the pullback connection in Eq. and the isometry condition Eq. But since the Levi-Civita connection is unique, this means that \begin{align} \label{eq:isometry-conns-agree} \varphi_{t}\text{ is an isometry of }g \qquad\Longrightarrow\qquad (\varphi_{t}^{*}\cd) = \cd \,. \end{align}

Since the two connections agree, we have the equality \begin{align} \cd_{X} T = \varphi_{-t}^{*} \left[ \cd_{\varphi_{t}^{*}(X)} \left( \varphi_{t}^{*} T \right) \right] \end{align} for all (the astute reader will have noticed that we have flipped the sign of in order to make later calculations prettier). Now we will find the infinitesimal version of this identity. Apply the inverse to both sides of this equality, differentiate both sides with respect to , and then evaluate at : \begin{align} \frac{d}{dt} \Big[ \varphi_{t}^{*} \cd_{X} T \Big]_{t=0} &= \frac{d}{dt} \Big[ \cd_{\varphi_{t}^{*}(X)} \left( \varphi_{t}^{*} T \right) \Big]_{t=0} \,, \\ \lie_{v} \left( \cd_{X} T \right) &= (\cd_{X} \lie_{v} T) + \cd_{(\lie_{v}X)} T \,. \end{align} This can be elucidated by expanding in abstract index notation (still suppressing indices on ), \begin{align} \lie_{v} \left( X^{a}\cd_{a} T \right) &= X^{a}(\cd_{a} \lie_{v} T) + (\lie_{v}X)^{a} \cd_{a} T \,, \\ (\lie_{v} X)^{a} \cd_{a} T + X^{a} \lie_{v} \cd_{a} T &= X^{a}(\cd_{a} \lie_{v} T) + (\lie_{v}X)^{a} \cd_{a} T \,. \end{align} Now cancel the terms on both sides. Noting that is arbitrary, we have derived an identity \begin{align} \lie_{v}\cd_{a} T = \cd_{a}\lie_{v} T \end{align} when is the Levi-Civita connection of and is a Killing vector field.

Note: another way to say this is

\begin{align} [ \lie_v, \cd_a ] T = 0 \,. \end{align}

In other words, when is a Killing vector field of the metric , commutes with the Levi-Civita connection of .

General connection coefficients for

Let us further study the difference and we will lift the restriction that is an isometry of First, we need some general properties of the difference of these connections. Much of this follows Sec. 3.1 of Wald.1

All connections agree when acting on scalar functions, so we have

\begin{align} \label{eq:diff-agree-scalar} (\varphi_{t}^{*}\cd)_{a}f-\cd_{a}f = 0 \end{align}

for all scalar fields . Next, examine their difference when acting on the product of a scalar field and a one-form field . Using the Leibniz rule, we find

\begin{align} \label{eq:diff-on-f-omega} (\varphi_{t}^{*}\cd)_{a}(f\omega_{b})-\cd_{a}(f\omega_{b}) = f \left[ (\varphi_{t}^{*}\cd)_{a}\omega_{b}-\cd_{a}\omega_{b} \right] \,. \end{align}

This tells us that the difference in fact only depends on the value of at each point, and not on the derivative of : consider a point where but the gradient of is arbitrarily large; still at that point, agrees with . There is a more rigorous argument given in 1, but the conclusion is the same: is just a linear transformation of its argument at each point on the manifold. Therefore, there is a tensor field (the connection coefficients) such that

\begin{align} \label{eq:diff-def-of-C} (\varphi_{t}^{*}\cd)_{a}\omega_{b}-\cd_{a}\omega_{b} = C(t)^{c}{}_{ab} \omega_{c} \,. \end{align}

This is extended by linearity to vectors and tensors in the usual way, with correcting each down index with a plus sign and each up index with a minus sign. For example, when acting on the metric, we have

\begin{align} \label{eq:diff-on-g} (\varphi_{t}^{*}\cd)_{a}g_{bc}-\cd_{a}g_{bc} = C(t)^{d}{}_{ab} g_{dc} + C(t)^{d}{}_{ac} g_{bd} \,. \end{align}

The first property of we can establish is that it is symmetric in its lower two indices. To show this, let the one-form be the gradient of a scalar, . Then we have

\begin{align} \label{eq:C-is-symm} (\varphi_{t}^{*}\cd)_{a}(\varphi_{t}^{*}\cd)_{b} f - \cd_{a} \cd_{b} f = C(t)^{c}{}_{ab} \cd_{c} f \,. \end{align}

Since is (by assumption) torsion-free, then so is . Therefore, both terms on the left hand side of Eq. \eqref{eq:C-is-symm} are symmetric, and so is their difference. Thus is symmetric in its lower two indices.

Now to make progress, we will expand as a power series in ,

\begin{align} \label{eq:C-power-series-first-few} C(t)^{c}{}_{ab} &= C^{(0)c}{}_{ab} + t C^{(1)c}{}_{ab} + \ldots \\ \label{eq:C-power-series} C(t)^{c}{}_{ab} &= \sum_{k=0}^{\infty} \frac{t^{k}}{k!}C^{(k)c}{}_{ab} \,, \\ C^{(k)c}{}_{ab} &\equiv \frac{d^{k}}{dt^{k}} C(t)^{c}{}_{ab} \Big|_{t=0} \,. \end{align}

Clearly since is the identity map, so

Now order-by-order, the connection coefficients’ expansion can be computed by taking derivatives of Eq. \eqref{eq:diff-on-g} (with contracted into the slot). We will perform the expansion for :

\begin{align} \frac{d}{dt} \left[ (\varphi_{t}^{*}\cd)_{X}g_{bc} \right] &= \frac{d}{dt} \left[ X^{a}C(t){}_{cab} + X^{a}C(t){}_{bac} \right] \\ \frac{d}{dt} \left[ \varphi_{t}^{*} \left( \cd_{(\varphi_{-t}^{*}X)} \varphi_{-t}^{*} g_{bc} \right) \right] &= X^{a} \left( C^{(1)}_{cab} + C^{(1)}_{bac} \right) \,. \end{align}

There are three terms on the left, one from each of the pullbacks. These are

\begin{align} \lie_{v}( \cd_{X} g_{bc}) + \cd_{(\lie_{-v}X)} g_{bc} + \cd_{X} \lie_{-v} g_{bc} &= X^{a} \left( C^{(1)}_{cab} + C^{(1)}_{bac} \right) \\ - X^{a} \cd_{a} \lie_{v} g_{bc} &= X^{a} \left( C^{(1)}_{cab} + C^{(1)}_{bac} \right) \\ \label{eq:cd-lie-g-is-C-plus-C} - \cd_{a} \lie_{v} g_{bc} &= C^{(1)}_{cab} + C^{(1)}_{bac} \,. \end{align}

Now we can do the following index gymnastics. Take Eq. \eqref{eq:cd-lie-g-is-C-plus-C} with the given index order, on the left hand side, ; add to it the same expression, but with the indices in the order ; and then subtract from it the same expression but with the indices in the order . Then using the symmetry on the last two indices of , we can solve for

\begin{align} \label{eq:C1-down-sol} C^{(1)}_{cab} = \frac{-1}{2} \Big( \cd_{a} \lie_{v} g_{bc} + \cd_{b} \lie_{v} g_{ac} - \cd_{c} \lie_{v} g_{ab} \Big) \,. \end{align}

Note: Again if is a Killing vector field of , this immediately vanishes. This says that to linear order in , agrees with .

Expansion of pullback connection

Here we are going to present an explicit formula in terms of Lie derivatives for the expansion of the pullback connection. I think this is not completely rigorous, but it seems satisfactory to me, and I checked it up to high order in Mathematica.

First, let us re-write the definition of the pullback connection Eq. \eqref{eq:pullback-connection-defn} as

\begin{align} (\varphi_{t}^{*}\cd)_{X} T &{}\equiv \varphi_{t}^{*} \left[ \cd_{d\varphi_{t}(X)} \left( \varphi_{t*} T \right) \right] \\ &{}= \varphi_{t}^{*} \left[ (\varphi_{-t}^{*} X)^{a} \cd_{a} \left( \varphi_{-t}^{*} T \right) \right] \,. \end{align}

Here, if the left hand side is evaluated at point , then on the right hand side, the up/down indices are in the spaces and . Now we are about to do something which a classical differential geometer might find illegal, but it seems fine to me. Recall that the pullback/pushforward commute with tensor product, so . We apply this to the right hand side, yielding

\begin{align} (\varphi_{t}^{*}\cd)_{X} T &{}= X^{a} \varphi_{t}^{*} \left[ \cd_{a} \left( \varphi_{-t}^{*} T \right) \right] \,. \end{align}

There is no pullback/pushforward acting on on the RHS because the pullback and pushforward have inverted each other. Now this notation looks quite bad, because the up index on is in . Inside the square brackets of the pullback, the down index on is very much in the space . However, this index is then pulled back to live in , so it may be correctly contracted with . I’m not sure if this is just a shortcoming of the abstract index notation, or if there is a deeper issue to understand here.

We continue being cavalier physicist and replace with . Therefore, we have an expression for the pullback connection as

\begin{align} \label{eq:pullback-conn-as-exp-cd-exp} (\varphi_{t}^{*}\cd)_{a} T = \exp[t\lie_{v}] \cd_{a}\exp[-t\lie_{v}] T \,. \end{align}

Through a standard formula in the theory of Lie algebras, we can formally re-express this as

\begin{align} (\varphi_{t}^{*}\cd)_{a} T ={}& (e^{t[\lie_{v},-]} \cd_{a}) T = (e^{t \ad_{\lie_{v}} } \cd_{a}) T \\ (\varphi_{t}^{*}\cd)_{a} T ={}& \sum_{k=0}^{\infty} \frac{t^{k}}{k!} \left((\ad_{\lie_{v}})^{k} \cd_{a}\right) T \,, \end{align}

where is the commutator, is the adjoint action of on , and

\begin{align} \label{eq:ad-k-cd} (\ad_{\lie_{v}})^{k} \cd_{c} = \underbrace{[\lie_{v},[\lie_{v},\cdots [\lie_{v}}_{ k\text{ Lie derivative commutators}} , \cd_{c} ]]] \,. \end{align}

Through explicit calculation in the xTensor package for Mathematica, I have confirmed that this expression agrees with the geometric definition up to 12th order in (this calculation takes only half a second for 6th order, but scales very badly with order because of two pushforwards and one pullback; and thus 12th order takes almost 7 minutes on my laptop).

This calculation allows us to find an explicit expression for as defined in Eq. \eqref{eq:C-power-series}. We follow the same approach as above, by acting with on . Now each side is expanded in a power series and powers of are equated. Again we play the index gymnastics game and take the combination of equations with indices in the orders . This allows us to find

\begin{align} \label{eq:C-k-explicit} 2 C^{(k)}_{cab} = \left((\ad_{\lie_{v}})^{k} \cd_{a}\right) g_{bc} + \left((\ad_{\lie_{v}})^{k} \cd_{b}\right) g_{ac} - \left((\ad_{\lie_{v}})^{k} \cd_{c}\right) g_{ab} \,. \end{align}

Note: Again inspect the case when is a Killing vector field of . Since , we recover the fact that and agree to all orders of .