Understanding diffeomorphism concept.

Question

I understood why homeomorphism is isomorphism in category of topological spaces. Because the structure we are interested about topological space is open set and the way homeomorphism is defined i.e. it is bijective bicontinuous exactly says that open set in one space exactly corresponds to open set in other space and vice versa. This makes much sense to me why the way homeomorphism is defined to be isomorphism in topological space category. Also, for metric spaces, the structure we care is distance and that's why it makes sense why isometry is isomorphism in category of metric spaces, because the way isometry is defined, it says the distance between two points in one space is equal to the distance between corresponding points in other spaces and vice versa. But I am not much clear about the diffeomorphism concept. Means why the way diffeomorphism is defined, gives us isomorphism in category of smooth manifolds. What is the structure we care about smooth manifolds and why it gets preserved the way diffeomorphism is defined.

Answer 1

The structures that are preserved by diffeomorphism are the various concepts of differentiability.

Here are some examples. Let $f : M \to N$ be a diffeomorphism of $n$-dimensional smooth manifolds.

Theorem: For any open subset of any Euclidean space $U \subset \mathbb R^n$ and for any function $g : U \to M$, the function $g$ is smooth if and only if the function $f \circ g : U \to N$ is smooth.

Theorem: For any function $h : N \to \mathbb R^n$, the function $h$ is smooth if and only if the function $h \circ f : M \to \mathbb R^n$ is smooth.

There are deeper examples as well. One of the key concepts in differentiable topology is the concept of the tangent space $TM$ of an $n$-dimensional smooth manifold $M$, which assigns to each point $x \in M$ and $n$-dimensional vector space denoted $T_x M$. For purposes of this answer I don't care about how these things are defined; what I care about is that a diffeomorphism "preserves" these things.

Theorem: For any diffeomorphism $f : M \to N$ of $n$-dimensional smooth manifolds and any $x \in M$ there is an induced linear isomorphism $D_x f : T_x M \to T_{f(x)} N$.

Theorem (the chain rule (restricted to diffeomorphisms)): For any diffeomorphisms $f : M \to N$ and $g : N \to P$ of $n$-dimensional smooth manifolds, and for any $x \in M$, we have $$D_x (g \circ f) = D_{f(x)} g \circ D_x f $$

One could go on and on. Just about any concept that arises in multivariable calculus will have an analogue on smooth manifolds. The defining concept of differential geometry (the study of curvature etc.) is a Riemannian metric on a smooth manifold $M$, which means a positive definite inner product defined each tangent space $T_x M$ that "varies smoothly" as the point $x$ varies; thus, for each $x \in M$ and each $v,w \in T_x M$ the inner product associates the real number $\langle v,w \rangle$. Again, I don't care about the precise definition; what I care about is that a diffeomorphism "preserves" these things:

Theorem: Given a diffeomorphism of smooth manifolds $f : M \to N$, and given a Riemannian metric on $M$ denoted $\langle v,w \rangle$ for each $v,w \in T_x M$, the following formula, for each $y \in N$ and each $u,v \in T_y N$, defines a Riemannian metric on $N$: letting $x = f^{-1}(y)$, $$\langle u,v \rangle = \langle (D_xf)^{-1}(u), (D_xf)^{-1}(v) \rangle $$

Finally, you ask about "why" these things are preserved by diffeomorphism, and I'll just address that for the first two theorems: the proofs of those theorems are exercises starting from the definitions of smooth manifolds and of diffeomorphisms.

Answer 2

A general definition for isomorphism in a category $C$ is a morphism $f:X\to Y$ with a double-sided inverse $g:Y\to X$, that is $$ f\circ g = 1_Y\quad g\circ f = 1_X. $$ The category of smooth manifolds has differentiable functions as morphisms. In this case, an isomorphism as defined above is precisely a diffeomorphism.