This question is cross-posted at the physics stack exchange at https://physics.stackexchange.com/q/488358/83357

Let $(\Sigma, q)$ be a non-degenerate submanifold of a Lorentzian manifold $(M,g)$. Let $N$ be the section of $T\Sigma ^g$. Physicists often talk about the evolution of $q$ along $N$ as $\mathcal{L}_Nq$. But this expression makes no sense as $N$ does not belong to $\mathfrak{X}(\Sigma)$. As such, Lie derivatives are defined using flows of vector fields; I don't see any natural way of extending it to arbitrary vector bundles$^{[1]}$.

What is happening here? What do physicists mean when they construct quantities like these$^{[2]}$? Even a link to a reference that treats this on a mathematically justifiable level is welcome.

[1] Naively, I would even expect that one would require some sort of a connection on the vector bundle to make this question tractable.

[2] The only argument I can think of is that the operation is actually being performed on the ambient manifold. Say $tan:\mathfrak{X}(M)\to \mathfrak{X}(\Sigma)$ is the canonical projection operation associated to the embedding ($tan:=q^{\sharp}\circ\iota \circ g^\flat$). Now, $tan^*(q)\in \Omega^2(M)$, so $\mathcal{L}_N(tan^*(q))\in\Omega^2(M)$ is well defined. But this feels like an incomplete picture, and possibly even wrong.


I am no physicist, but I have thought about connections associated to submanifolds. I don't know how to describe the specific construction in your question, but it reminds me of the Bott connection associated to the foliation. When $M$ is pseudo-Riemannian and $\Sigma$ is nondegenerate, $\Sigma$ is locally the leaf of a foliation via the Riemannian structure. So, I will just deal with foliations $\mathcal{F}$ where $\Sigma$ is a leaf.

Connections associated to a foliation:

There is the Bott connection on the normal bundle of $\mathcal{F}$. The normal bundle $N_{M/\mathcal{F}}$ is defined in the same way as the normal bundle of a submanifold, but is on $M$: $$N_{M/\mathcal{F}} = T_M/T_\mathcal{F}.$$ On the leaves of the foliation, the normal bundle has a canonical connection given by the Lie bracket: $$\nabla^{Bott}: N_{M/\mathcal{F}} \otimes \Gamma(T_\mathcal{F})\to N_{M/\mathcal{F}},\\ \nabla^{Bott}_X Y = [X,Y] = \mathcal{L}_X(Y).$$ It is a nice exercise to check that this gives a well-defined connection. This restricts to a connection on $N_{M/\mathcal{F}}|_\Sigma$. To compute $\nabla_X Y$ for $X \in \Gamma(T_\Sigma)$, first extend $X$ to $\tilde X$ tangent to the foliation in a neighborhood of $\Sigma$, extend $Y$, and then compute with the Bott connection. There is also a dual connection on the conormal bundle $N^\vee_{M/\mathcal{F}} = (T_\mathcal{F}^\perp \subseteq \Omega^1_M)$ over the foliation, again given by the Lie derivative.

I suspect that your object $\mathcal{L}_Nq$ is defined similarly, but using the orthogonal complement under the Riemannian structure instead of the normal or conormal bundles. It should be well-defined in the same way the Bott connection is well-defined.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.