I'm currently reading Lee's Introduction to Smooth Manifolds, in which he makes a remark along the following lines: "The next corollary can be viewed as a more invariant version of the rank theorem." My question isn't about the corollary or the rank theorem itself, but about Lee's use of the word "invariant." This word is thrown around a lot in the community, and I have always seen it used as a noun, meaning a property left unchanged by a transformation. But here it's an adjective, and I'm having trouble figuring out what Lee is trying to communicate. What does it mean for a statement to be invariant? Does it relate to the noun meaning I'm familiar with?
For reference, here's the theorem and the corollary:
Rank Theorem: Suppose $M$ and $N$ are smooth manifolds of dimensions $m$ and $n$, respectively, and $F : M \to N$ is a smooth map with constant rank $r$. For each $p \in M$ there exist smooth charts $(U,\phi)$ for $M$ centered at $p$ and $(V,\phi)$ for $N$ centered at $F(p)$ such that $F(U) \subset V$, in which $F$ has a coordinate representation of the form $\hat{F}(x^1,\ldots,x^r,x^{r+1},\ldots,x^m) = (x^1,\ldots,x^r,0,\ldots,0)$.
Corollary: Let $M$ and $N$ be smooth manifolds, let $F : M \to N$ be a smooth map, and suppose $M$ is connected. Then the following are equivalent: (a) For each $p \in M$ there exist smooth charts containing $p$ and $F(p)$ in which the coordinate representation of $F$ is linear. (b) $F$ has constant rank.