Is the differential at a regular point, a vector space isomorphism of tangent spaces, also a diffeomorphism of tangent spaces as manifolds

Question

Note: My question is not "If $f$ is a diffeomorphism, then is the differential $D_qf$ an isomorphism?"

My book is From Calculus to Cohomology by Ib Madsen and Jørgen Tornehave. I didn't study much of the definitions or theorems in the book, if they were already found in An Introduction to Manifolds by Loring W. Tu. I mostly assume they're the same until there is evidence otherwise.

In Chapter 11, Madsen and Tornehave define "local index", which looks to me like just a different way to say sign of the determinant of the Jacobian matrix that represents the differential (See Tu Proposition 8.11; Tu Section 23.3; Madsen and Tornehave Lemma 10.1; Madsen and Tornehave Lemma 10.3; Wikipedia Degree of a continuous mapping, specifically this).

Now, for a regular point $q \in f^{-1}(p)$ for a regular value $p$ that is in the image of $f$ (For a regular value $p$ that isn't in the image of $f$, I'm sure there are neat vacuous arguments that I'm gonna skip), it says the local index is defined as $1$ if $D_qf$ preserves orientation and $-1$ otherwise.

I was surprised to see orientation-preserving as an adjective for an isomorphism of vector spaces because I'm used to seeing orientation-preserving as an adjective for diffeomorphisms of manifolds. However, $T_pN^n \cong \mathbb R^n$ (vector space isomorphic), so I guess tangent spaces of manifolds are manifolds as well, assuming the image of an oriented manifold under a vector space isomorphism is also an oriented manifold or something.

• (This question seems to confirm that tangent spaces of manifolds are manifolds, although I think the definition in the question is the same as the one in Madsen and Tornehave but different from the one in Tu). Actually, upon a second reading of the answer of Alex Mathers to that question, I think I have an answer to my question: Any vector space isomorphism, of tangent spaces of manifolds or any other vector spaces, turns out to be a homeomorphism. While my question is diffeomorphism, it turns out John M. Lee's Example 1.24, which was pointed out by Alex Mathers, shows that any isomorphism of finite real vector spaces is a diffeomorphism as well. Rather than analyzing the example, I'm going to try a different proof.)

I think that $D_qf$, or $f_{*, q}$ in Tu's notation, is a diffeomorphism of the tangent spaces as manifolds because:

1. $D_qf$ is surjective either by definition of $q$ being a regular point (Tu Definition 8.22) or by $q \in f^{-1}(p)$ and definition of $p$ being regular value of $f$ that is in the image of $f$ (Madsen and Tornehave Chapter 11).

2. $D_qf$ is a homomorphism of tangent spaces (almost immediately from definition, but anyway, this follows from Tu Exercise 8.3).

3. $D_qf$ is injective, by this, because of (1), (2) and that the dimensions of $T_qN$ and $TpM$ are finite and equal.

4. $D_qf$ is a local diffeomorphism of manifolds if and only if for each $X_q \in T_qN$, the (double) differential $D_{X_q}(D_qf): T_{X_q}(T_qN) \to T_{D_qf(X_q)}(T_pM)$ is an isomorphism of (double) tangent spaces, by the Inverse Function Theorem for manifolds (specifically by Tu Remark 8.12, which gives a "coordinate-free description" for Tu Inverse Function Theorem for manifolds (Tu Theorem 6.26))

5. $D_qf$ is a diffeomorphism of manifolds if and only if $D_qf$ is a bijective local diffeomorphism of manifolds (at each $X_q \in T_qN$) by this.

6. $D_qf$ is an isomorphism of tangent spaces by (1), (2) and (3).

7. Every $D_{X_q}(D_qf)$ is identical to $D_qf$ itself, by Tu Problem 8.2 (also found in this question and this question), because of (2).

8. Every $D_{X_q}(D_qf)$ is an isomorphism of tangent spaces because of (6) and (7).

9. $D_qf$ is a local diffeomorphism of manifolds (at each $X_q \in T_qN$) by (4) and (8).

10. $D_qf$ is a diffeomorphism of manifolds by (1), (3), (5), and (9).

Answer 1

The answer to your question is yes but, at least according to most treatments I know, you don't really need to know the answer to make sense of the definition of the local index. This is because the authors likely refer to the concept of "orientation-preserving" isomorphisms of oriented vector spaces from algebra rather than the "orientation-preserving" for diffeomorphisms of manifolds from geometry. The latter definition involves smoothness while the former definition doesn't. As it turns out $D_qf$ is orientation-preserving as a vector space isomorphism if and only if $D_qf$ is orientation-preserving as a diffeomorphism of manifolds, but you need an interpretation of how a vector space becomes a manifold.

To make your argument precise, the first question you need to ask yourself is how do you want to think of $T_qN$ (and $T_pM$) as a manifold? That is, what is the topology and the smooth structure on $T_qN$? Without answering this question, you can't really argue that $D_qf$ is a homeomorphism/diffeomorphism. There are at least two choices which make sense:

1. Think of $T_qN$ as a vector space. Any vector space $V$ has a unique smooth structure which is obtained by declaring some isomorphism $\psi \colon \mathbb{R}^n \rightarrow V$ to be a global chart for $V$. You can check that the smooth structure doesn't depend on the choice of the isomorphism and, once you use one isomorphism, any other isomorphism will also be a global chart. If you endow two vector spaces $V,W$ with the natural smooth structures described above, you can check that any linear map $S \colon V \rightarrow W$ will automatically be smooth (in particular, continuous). Hence, if $S$ is bijective, it will be a diffeomorphism (as $S^{-1}$ is also linear, hence smooth). You can also use the fact that the differential of $S$ can be identified with $S$ itself, but it just complicates the argument. In particular, if you apply this argument to $V = T_qN, W = T_pM$ and $S = D_qf$, you'll get that $D_qf$ is a diffeomorphism.
2. Think of $T_qN$ as a submanifold of the tangent bundle $TN$. One can check that $T_qN$ is indeed an embedded submanifold of $TM$ so it has a natural unique smooth structure compatible with the subspace topology, which, as it happens, turns out to be the same structure you would get if you used the vector space structure. With this interpretation, you can check that $D_qf$ is a diffeomorphism by using slice charts around $T_qN$ and $T_pM$ (which come from the construction of charts on $TN,TM$) and verifying that, in local coordinates, $D_qf$ is linear bijective map, hence a diffeomorphism. You can also argue in various other ways.

Next, in order to make sense of your interpretation, note that it is not enough to give $T_qN$ the structure of a manifold. You need to orient it as well. How you will do that depends on your definition of orientation (as there are many equivalent definitions). If an orientation is defined by giving an oriented atlas, the easiest thing to do is to work with the first interpretation above. If $X \colon U \rightarrow N$ is an oriented chart around $q$ with $X(a) = q$, define an oriented smooth structure on $T_qN$ by declaring the differential $DX|_a \colon T_a(\mathbb{R}^n) \rightarrow T_qN$ to be an oriented chart (where you identify $T_a(\mathbb{R}^n)$ with $\mathbb{R}^n$ in the usual way). If your definition of orientation is different, you might need to do something different.

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As you can see, there are many details to fill in order to work with your interpretation. However, most books I know (I haven't checked Tu nor Marsden) also discuss the notion of an orientation of a vector space which is a pure linear algebra notion unrelated to any issues of smoothness. Then one defines when a map between oriented vector spaces is orientation preserving and finally, one shows that the definition of orientation on a manifold $N$ induces an orientation for each tangent space $T_qN$ (which "varies smoothly" with respect to $q$). Then, the definition of index is with respect to the notion of orientation preserving/reversing linear maps between oriented vectors spaces and not diffeomorphisms between oriented manifolds. This gives a conceptually cleaner treatment as it seperates the issue of smoothness from the issue of being orientation preserving/reversing.