Consider a manifold $M$ with a morse function $t$. For the regular points, $t^{-1}(c)$ are embeddings, and the homotopy changes as one jumps from one regular point to another. But, my question is: what can one say about the critical points, i.e. $t^{-1}(c)$ where $c$ is the critical point. Looking at examples, I see why this fails to be even a smooth submanifold. This raises the question, as to what we can say about it.
My intuition leads me to ask the following questions: Is it some sort of a stratification? If yes, is it well studied, and where can I find it? Also, if the ambient manifold has a metric structure, is it possible to use it in anyway to induce anything on the substructure?