If I describe some geometric shape (like a square in $\Bbb R^2$, or a torus in $\Bbb R^3$) and then change the basis which I use to describe it, does this have any relation to the invariance of tensors under basis transformations? I understand that a torus for instance doesn't have a set of components in the same way that a vector does, but the basis which is used to describe it doesn't influence its actual structure. Is there a term for this in the same way we call objects who's components transform predictably "tensors"?
I am studying physics not mathematics so forgive any egregious misunderstanding of geometry.