Let $(M^n,g)$ be a smooth compact Riemannian manifold. It is well known that any sequence $\{X_i\}$ of compact subsets of $M$ has a subsequence which converges in the Hausdorff metric to a compact subset $X\subset M$.

Assume now that $\{X_i\}$ are, in addition, smooth connected submanifolds with a uniform lower bound on the sectional curvature and a uniform upper bound on the diameter with respect to the induced intrinsic (!) metric.

**Question. What is known about a limit space $X$?**
E.g. should $X$ have an integer Hausdorff dimension?

Remark. Under the above assumptions, the Gromov compactness theorem implies that there is a subsequence converging to a compact Alexandrov space $Y$ in the Gromov-Hausdorff sense (thus $Y$ is not a subset of $M$ a priori). **Is there any relation between $X$ and $Y$?** (I believe there is a 1-Lipschitz map onto $Y\to X$.)