Question

Show that if \(\psi: M \rightarrow N\) is a local diffeomorphism at \(P \in M\), then \(\psi_{* P}: \mathcal{T}_{P}(M) \rightarrow \mathcal{T}_{\psi(P)}(N)\) is a vector space isomorphism.

Short answer

The vector space isomorphism has been proven: by stating that \(\psi_{* P}: \mathcal{T}_{P}(M) \rightarrow \mathcal{T}_{\psi(P)}(N)\) is bijective due to the local diffeomorphism of \(\psi\) and linear as inherited from their parent maps. Therefore, \(\psi_{* P}: \mathcal{T}_{P}(M) \rightarrow \mathcal{T}_{\psi(P)}(N)\) is a vector space isomorphism.

Step-by-step solution

Prove Bijectivity

Bijectivity of a function implies that it is both injective (or one-to-one) and surjective (or onto). Since \(\psi\) is a local diffeomorphism, it means that near every point \(P \in M\), it behaves like a diffeomorphism. Given that diffeomorphisms are bijective, \(\psi_{* P}: \mathcal{T}_{P}(M) \rightarrow \mathcal{T}_{\psi(P)}(N)\) is also be bijective. Thus, the map is injective and surjective.

Prove Linearity

Linearity implies that for vectors \(v, w \in \mathcal{T}_{P}(M)\) and scalars \(a, b \in \mathbb{R}\), \(\psi_{* P}(av + bw) = a\psi_{* P}(v) + b\psi_{* P}(w)\). This holds since \(\psi_{* P}\) is inherited through the local diffeomorphism from the already linear maps \(d \psi_{P}\) by definition. Therefore, \(\psi_{* P}\) is linear.

Finalize Proof

After showing that the map is bijective and linear, it can be concluded that \(\psi_{* P}: \mathcal{T}_{P}(M) \rightarrow \mathcal{T}_{\psi(P)}(N)\) is a vector space isomorphism.

Key concepts

Local Diffeomorphism

A local diffeomorphism is a special kind of function defined near a particular point that maintains the properties of a diffeomorphism. Simply put, around any point of interest on the manifold, the function behaves like an isomorphism between smooth spaces. This means that, locally, it has an inverse and preserves both structure and form.
- **Locally Isomorphic**: At any given point, a local diffeomorphism acts like a neat "copying" of a tiny patch of space onto another. If you zoom in closely enough around the point, you could actually map it smoothly to another space. - **Slightly Different than Global Diffeomorphisms**: Unlike a global diffeomorphism which maps entire manifolds in a similar fashion, a local diffeomorphism only works over small regions.
Understanding this concept helps us venture into proving why the map induced on tangent spaces by a local diffeomorphism is bijective, which is a crucial step in showing isomorphism in vector spaces.

Tangent Spaces

Tangent spaces are mathematical concepts that help describe how a curve behaves or what it might look like at a particular point. You can think of it like a plane or space that "just touches" the curve at one point and runs along it without piercing through. - **Vectors at a Point**: A tangent space at a point on the manifold is a collection of tangent vectors at that point. It's a "local" view—those vectors touch that point and indicate possible directions to move away from that point. - **Crucial to Calculus on Manifolds**: In manifold calculus, tangent spaces let us do operations similar to those in regular calculus, like differentiating functions and finding directions. When a local diffeomorphism acts on a manifold, it induces a map between tangent spaces at corresponding points. This map is linear and bijective, meaning it establishes an isomorphic relationship between the tangent spaces involved.

Linearity

Linearity is a key property when dealing with functions, especially in the context of vector spaces. It describes a function that satisfies two important principles: additivity and homogeneity.- **Additivity**: For linear maps, adding two inputs and then mapping the result is the same as individually mapping the inputs and then adding the results. Mathematically, for vectors \( v \)and \( w \), this is \( f(v+w) = f(v) + f(w) \).- **Homogeneity**: Scaling an input and then mapping it is the same as mapping the input first and then scaling the result. This means for a scalar \( a \) and a vector \( v \), \( f(av) = a f(v) \).In the context of our problem, showing that the map between tangent spaces is linear is crucial. That's because, combined with bijectivity, it confirms the map forms an isomorphism—allowing us to treat the two tangent spaces as structurally identical in terms of vector space properties.