Differentation under the integral sign (Leibniz integral rule, Reynolds transport theorem) in differential-geometric terms takes the form [1, 2]:
Let $\phi\colon \mathbb R\times M\to M$ be a 1-parameter family of diffeomorphisms $\phi_t = \phi(t, \cdot)$ and $\alpha\in \Omega^n(\mathbb R\times M)$ be a family of $n$-forms $\alpha_t\in \Omega^n(M)$. Assume that $\phi_0=\mathrm{id}_M$. If $V$ is an $n$-dimensional submanifold of $M$, then $$\frac{d}{dt}\int_{\phi_t(V)} \alpha_t = \int_{\phi_t(V)} \left(\frac{\partial}{\partial t} + \mathcal L_X\right)\alpha_t,$$ where $X(t, \phi_t(x)) = d\phi_t(x)/dt$ is a time-dependent velocity field on $M$. (Remarkably, although $\phi_0 = \mathrm{id}_M$, $\phi_t$ does not need to form a flow on $M$, i.e. does not need to be a representation of $\mathbb R$).
I am looking for a generalization of this formula, where the time axis $\mathbb R$ is replaced by a general manifold $N$ and $d/dt$ is replaced by a tangent vector $v\in T_pN$. Unfortunately, I don't see how to generalize the standard proof (in which one integrates the proposed field $X$ to get a flow $\theta_s$ on $\mathbb R\times M$, and then using the assumption $\phi_0 = \mathrm{id}_{M}$ retrieves $\phi_t(V)$ as projections of $\theta_s(0, V)$).
I wonder if you could suggest me appropriate references for this.
References
- T. Frankel, Geometry of Physics, Theorem 4.42.
- M. Reddiger, B. Poirer, On the Differentiation Lemma and the Reynolds Transport Theorem for Manifolds with Corners, arXiv:1906.03330.