I am reading about scissors congruence groups and subspaces of quadratic spaces have come up, but I'm not very familiar with these definitions and apologize in advance if something I wrote doesn't make sense. I would appreciate any intuition or references for my following questions.
First the setting I am working in:
Def: a geometry $X = (E, q)$ over the field k is a vector space $E$ equipped with a non-degenerate quadratic form $q$, and corresponding isometry group $I(X)$.
Def: A subspace $U$ of the quadratic form is a linear subspace of $E$ such that $q$ is non-degenerate.
My questions are:
In the case of the geometry defined by a quadratic form with signature $(n_-, n_+)$, I am wondering if specifying signature and dimension completely classifies the subspaces up to isometry?
I've seen someone refer to an isometric copy of a subspace, it was in the case of a hyperbolic or spherical geometry (so the equator inside $S^2$). However, if we have just a general quadratic form, what would this mean?
I am curious if there are conditions on the field? (though for the above questions, I would have happy to work in $\mathbb{R}$.