We know that $M_{m \times n } $ is isomorphic to $R^{mn}$. Let's take a smooth manifold $\mathbb{M}$ in $R^{mn}$ and fix a point in $R^{n}$, say, $p$. Now realize the manifold $\mathbb{M}$ as a subset of $M_{m \times n}$. Let each matrix in the subset so realized act on $p$, and we get a subset of $R^{m}$. So, what can be said about this subset of $R^{m}$. Is this always a manifold? How does it's topological properties depend on $p$ and the initially chosen smooth manifold $\mathbb{M}$, can we find some invariant topological properties which are preserved under some conditions from $\mathbb{M}$ and the newly found subset of $R^{m}$

Background

I read Loring Tu's "Introduction to smooth manifolds" for my MS thesis. " but I hadn't covered Cohomology of smooth manifolds well. I am not able to "use" the theory I read so far. Would you suggest a way to think about the above problem? Some sort of hint to start with?

Currently, I am not enrolled in a PhD program, for a year, I have to do some project or independent study, I was thinking of delving into the above problem deeply and maybe publishing.

Any references or any suggestions in general will be very much appreciated.