Smooth structure on a quotient vector space

Question

How do I know if $$f:\mathbb{R}^n/(\mathbb{R}^k\times \{\mathbf{0}^{n-k}\})\to \mathbb{R}^n/(\mathbb{R}^k\times \{\mathbf{0}^{n-k}\})$$ is smooth? Can't find the definition of the canonical smooth structure on a quotient vector space. (Context: Trying to show that a union of quotient fibers can be made into a smooth vector bundle. This is a transition map...)

Answer 1

The definition can be formulated in several equivalent ways, but one is that a function $f:\mathbb{R}^n/\mathbb{R}^k \to \mathbb{R}^n/\mathbb{R}^k$ is smooth if and only if the corresponding function $F:\mathbb{R}^n\to\mathbb{R}^n$ is smooth, where $F$ is defined by $F(\mathbf{x}) = f(\mathbf{x}+\mathbb{R}^k)$ (i.e. $F$ is the function that is constant on $\mathbb{R}^k$-cosets and its value is the value of $f$ on that coset).

Alternatively, you can choose an isomorphism $\mathbb{R}^n/\mathbb{R}^k \to \mathbb{R}^{n-k}$ and then $f$ is smooth if and only if its composition with this isomorphism is a smooth function $\mathbb{R}^{n-k}\to\mathbb{R}^{n-k}$. I think your notation means the subspace $\mathbb{R}^k$ is the subspace you get by setting the last $n-k$ coordinates equal to $0$, in which case you can find such an isomorphism by simply projecting onto the last $n-k$ coordinates.

However, given the context of the problem, don't you want the transition maps to be linear on the fibers, not just smooth? More carefully: if $U_1$ and $U_2$ are subsets of the base space over which the vector bundle is trivial and such that $U_1\cap U_2 \neq\emptyset$, for your original rank $n$ bundle, you get a smooth transition map $g_{12}:U_1\cap U_2\to \operatorname{GL}_n(\mathbb{R})$, and you want to show that this induces a smooth map $U_1\cap U_2\to \operatorname{GL}_{n-k}(\mathbb{R})$. Then for each $x \in U_1\cap U_2$, you need to show that the $g_{12}(x) \in\operatorname{GL}_n(\mathbb{R})$ induces an injective linear map $\mathbb{R}^n/\mathbb{R}^k\to\mathbb{R}^n/\mathbb{R}^k$, so that you get an element of $\operatorname{GL}_{n-k}(\mathbb{R})$. Then you want to show that the resulting map $U_1\cap U_2\to \operatorname{GL}_{n-k}(\mathbb{R})$ is smooth.