# Is the Lie derivative along the normal well defined?

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.
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.