My book is Connections, Curvature, and Characteristic Classes by Loring W. Tu (I'll call this Volume 3), a sequel to both Differential Forms in Algebraic Topology by Loring W. Tu and Raoul Bott (Volume 2) and An Introduction to Manifolds by Loring W. Tu (Volume 1).
I refer to Section 27.2 and to my previous question: Manifold structure such that a bijection of sets becomes a diffeomorphism of manifolds? Group structure for group isomorphism?.
Question A: The following is my understanding of the part "Using the bijection...frame manifold of the vector space $V$". Is this correct?
Note: I ask about frame manifold not frame bundle. $V$ is still finite $\mathbb R$-vector space of dimension $r$ and not a smooth $\mathbb R$-vector bundle of rank $r$.
- For smooth manifolds $(M,\mathscr A)$ and $(M,\mathscr B)$ with the same underlying set $M$ and respective smooth atlases $\mathscr A$ and $\mathscr B$, it's possible that $(M,\mathscr A)$ and $(M,\mathscr B)$ are diffeomorphic but neither of the identity maps $id_{M,\mathscr A,\mathscr B}:$ $ (M,\mathscr A)$ $\to$ $(M,\mathscr B)$ and $id_{M,\mathscr B,\mathscr A}:$ $ (M,\mathscr B)$ $\to$ $(M,\mathscr A)$ are diffeomorphisms.
- 1.1 I think this is what Problem 6.1 of Volume 1 (also the example here) is an example of. For reference, here's a solution given by Richard G. Ligo.
We have a right, free and transitive (regular) action $\mu: Fr(V) \times GL(r, \mathbb R) \to Fr(V)$. For each $u \in Fr(V)$, we can define a map $\mu_u: GL(r, \mathbb R) \to Fr(V)$, $\mu_u(g) := \mu(u,g)$ that turns out to be a bijection between $Fr(V)$ and the manifold (and Lie group) $GL(r, \mathbb R)$
Let $v,w \in Fr(v)$. We consider $Fr_v(V) = (Fr(V),\mathscr A_v)$ and $Fr_w(V) = (Fr(V),\mathscr A_w)$ be smooth manifolds with the same underlying set $Fr(V)$ with $\mathscr A_v$ and $\mathscr A_w$ respectively uniquely determined by the bijections of sets $\mu_v: GL(r,\mathbb R) \to Fr(V)$ and $\mu_w: GL(r,\mathbb R) \to Fr(V)$ becoming (smooth) diffeomorphisms of (smooth) manifolds $\mu_v^d: GL(r,\mathbb R) \to Fr_v(V)$ and $\mu_w^d: GL(r,\mathbb R) \to Fr_w(V)$.
If each $Fr_v(V)$ and $Fr_w(V)$ are diffeomorphic, then it is already well-defined to say that we can make the set $Fr(V)$ into a (smooth) manifold called the frame manifold of $V$ by choosing the manifold that $Fr(V)$ becomes as $Fr_u(V)$ for any $u \in Fr(V)$, in the sense that even though we might have $Fr_v(V) \ne Fr_w(V)$, we have at least that $Fr_v(V)$ and $Fr_w(V)$ are diffeomorphic. So, the frame manifold is well-defined "up to diffeomorphism" or something.
We can show $Fr_v(V)$ and $Fr_w(V)$ are diffeomorphic by either $\mu_v^d \circ (\mu_w^d)^{-1}$ or its inverse.
Additionally, we can show that $id_{Fr(V),\mathscr A_v,\mathscr A_w}: (Fr(V),\mathscr A_v) \to (Fr(V),\mathscr A_w)$ and $id_{Fr(V),\mathscr A_w,\mathscr A_v}: (Fr(V),\mathscr A_w) \to (Fr(V),\mathscr A_v)$ are diffeomorphisms by noting $\mu_w = \mu_v \circ l_a$, where $a$ is the unique element of $GL(r,\mathbb R)$ such that $w=\mu(v,a)$.
- 6.1. Thus, left multiplication enables us to prove something stronger than that $Fr_v(V)$ and $Fr_w(V)$ are diffeomorphic: That $Fr_v(V)$ and $Fr_w(V)$ are diffeomorphic by either of the identity maps $id_{Fr(V),\mathscr A_v,\mathscr A_w}$ and $id_{Fr(V),\mathscr A_w,\mathscr A_v}$. Therefore, the frame manifold is well-defined in an even stronger sense than in (4) because $Fr_v(V)$ and $Fr_w(V)$ are "identical" in the sense of Guess 1 or Guess 2 below.
Question B: (Assuming (1) is correct). What's the term, if any, to describe the manifolds $(M,\mathscr A)$ and $(M,\mathscr B)$ as diffeomorphic by the identity map, or alternatively what can we say about such $(M,\mathscr A)$ and $(M,\mathscr B)$?
Guess 1: There is a term, and it is that they are "identical" in the sense that they are not "distinct", where "distinct" is in the sense of Problem 6.1b of Volume 1. This is because we can say that the maximal (smooth) atlases $\mathscr M(\mathscr A)$ and $\mathscr M(\mathscr B)$ are equal (Note: The "$M$" of "$\mathscr M$" stands for "maximal" and does not refer to the underlying set $M$.). Thus, what Tu is trying to say is that $Fr_v(V)$ and $Fr_w(V)$ are not only diffeomorphic but actually "identical" i.e. $\mathscr M(\mathscr A_v)$ $=$ $\mathscr M(\mathscr A_w)$.
Guess 2: It is not true in general that we can say that $\mathscr M(\mathscr A) = \mathscr M(\mathscr B)$. However, we can say that, for the specific case of $M=Fr(V)$, $\mathscr A = \mathscr A_v$ and $\mathscr B = \mathscr A_w$, we have that $\mathscr M(\mathscr A_v)$ $=$ $\mathscr M(\mathscr A_w)$. This is the same sense as in Guess 1 that $Fr_v(V)$ and $Fr_w(V)$ are "identical".
Question C: (I happened to notice Problem 6.3 of Volume 1 as I was looking up Problem 6.1 of Volume 1) Is Problem 6.3 of Volume 1 precisely proving the frame manifold of Volume 3 is well-defined?
- Solution at the back of the book (which says the maximal atlases are the same): Solution to Problem 6.3
