Topological Properties of Subsets of $R^{m}$ induced by Smooth Manifolds in Matrix Spaces

Question

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 $R^{m}$

Some idea We can think of the manifold acting on $p$ as a linear map associated with $p$ acting on the manifold and mapping it to a lower-dimensional Euclidean space. So in essence what we have is a linear mapping of a manifold, and we are interested in investigation of is there anything interesting happening when we change the linear map induced by $p$ or when we alter the choice of manifold $M$.

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.

Answer 1

Let me write $\mathbb M\cdot p$ for the set $\{Mp,M\in\mathbb M\}\subset\mathbb R^m$. Clearly it is a linear image of a manifold, which already gives you some amount of information. However, it may not be a manifold; for instance, if $\gamma$ is any smooth curve in the plane, then the set of $2\times2$ matrices of the form $$\begin{pmatrix}\gamma_1(t) & t \\ \gamma_2(t) & 0 \end{pmatrix}$$ is a manifold (as the image of a proper, smooth, injective immersion). However, for this $\mathbb M$ and $p=(1,0)$, we clearly have $\mathbb M\cdot p=\operatorname{im}\gamma$, so it is a manifold if and only if the image of $\gamma$ is, which may not be the case (say for $\gamma(t)=(1-t^2,t^3-t)$ for instance).