Explore why the differential of a diffeomorphism must be a linear isomorphism between tangent spaces at corresponding points | Step-by-Step Solution

Understanding Why Diffeomorphisms Have Isomorphic Differentials 🎯

What We're Solving:

We need to understand why when we have a diffeomorphism (a smooth, invertible map between manifolds), its differential must be a linear isomorphism between the tangent spaces at corresponding points.

The Approach:

This is a "smoothness preservation" story! We'll use the chain rule and the fact that diffeomorphisms have smooth inverses to show that the differential must be invertible. The key insight is that if f is a diffeomorphism, then f ∘ f⁻¹ = identity, and this relationship carries over to the differential level.

Step-by-Step Solution:

Step 1: Set up what we know

• Let f: M → N be a diffeomorphism between manifolds M and N
• This means f is smooth, bijective, and f⁻¹ is also smooth
• At any point p ∈ M, we have the differential df_p: T_p(M) → T_{f(p)}(N)

Step 2: Use the inverse relationship Since f is a diffeomorphism, we know that:

• f ∘ f⁻¹ = id_N (identity on N)
• f⁻¹ ∘ f = id_M (identity on M)

Step 3: Apply the chain rule to differentials Taking differentials of these compositions:

• d(f ∘ f⁻¹)_{f(p)} = d(id_N)_{f(p)}
• d(f⁻¹ ∘ f)_p = d(id_M)_p

By the chain rule:

• df_{f⁻¹(f(p))} ∘ d(f⁻¹)_{f(p)} = I (identity on T_{f(p)}(N))
• d(f⁻¹)_{f(p)} ∘ df_p = I (identity on T_p(M))

Step 4: Recognize what this means Since df_p = df_{f⁻¹(f(p))}, we have:

• df_p ∘ d(f⁻¹)_{f(p)} = I
• d(f⁻¹)_{f(p)} ∘ df_p = I

This shows that d(f⁻¹)_{f(p)} is both a left and right inverse to df_p!

Step 5: Conclude the result Since df_p has an inverse (namely d(f⁻¹)_{f(p)}), it must be:

• Injective (one-to-one)
• Surjective (onto)
• Therefore, a linear isomorphism!

The Answer:

The differential of a diffeomorphism must be a linear isomorphism because:

• 1. The existence of a smooth inverse f⁻¹ guarantees that df_p has an inverse
• 2. The chain rule applied to f ∘ f⁻¹ = identity shows that d(f⁻¹)_{f(p)} is the inverse of df_p
• 3. A linear map with an inverse is automatically a linear isomorphism

Memory Tip:

Remember "Smooth in, smooth out, inverse throughout!" If you can go smoothly in both directions between manifolds, you can also go smoothly in both directions between their tangent spaces. The diffeomorphism property "lifts" from the manifold level to the tangent space level through the differential! 🔄

This is a beautiful example of how global properties (being a diffeomorphism) determine local linear algebraic properties (isomorphic tangent spaces).