Convergence of a sequence of manifolds

Question

Suppose $(M_k)_k$ is a sequence of connected Riemannian manifolds, and let $a_k$ be a function or a vector field defined on $M_k$. How can one conceive the limits

$M_k\to M_\infty$, $M_\infty$ another Riemannian manifold

$a_k\to a_\infty$, $a_\infty$ a function/vector field defined on $M_k$.

For instance, imagine a sequence of hemispheres of radius $k$. Intuitively they must converge in some sense to a flat plane. But in which sense?

Answer 1

You may use the concept of (smooth) Cheeger-Gromov convergence: I find the definition in p.129 in here:

Definition (Cheeger-Gromov Convergence in $C^\infty$). A sequence $\{(M_k, g_k, O_k)\}$ of complete pointed Riemannian manifolds converges to a complete pointed Riemannian manifold $(M_\infty, g_\infty, O_\infty)$ if there exists:

1. An exhaustion $U_k$ of $M_\infty$ with $O_\infty\in U_k$;

2. A sequence of diffeomorphisms $\phi_k : U_k\to V_k \subset M_k$ with $\phi(O_\infty) = O_k$ such that $\phi_k^* g_k$ converges in $C^\infty$ to $g_\infty$ on compact sets in $M_\infty$.

If you have a function/vector fields $a_k$ on $M_k$, you can also consider the sequence of functions $\phi_k^*a_k$ or vector fields $(\phi^{-1}_k)_* a_k$ on $M_\infty$, so you can reguire that $a_k$ converges locally smoothly to $a_\infty$.