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Chapter 15: Problem 10

Show that the tangent space \(T_{(\rho q)}(M \times N)\) at any pount \((p, q)\) of a product manifold \(M \times N\) is naturally isomorphic to the dircct sum of tangent spaces \(T_{p}(M) \oplus T_{q}(N)\)

Short Answer

Expert verified
The required natural isomorphism is therefore given by the defined map \(\Phi: T_{(\rho q)}(M \times N) \rightarrow T_{p}(M) \oplus T_{q}(N)\) where \(\Phi(X_p, Y_q) = (X_p, Y_q)\).

Step by step solution

01

Define Tangent Spaces at each point in M and N

First, you need to define the tangent spaces at a point \(p \in M\) and a point \(q \in N\). These are denoted by \(T_{p}(M)\) and \(T_{q}(N)\) respectively. They consist of all tangent vectors at point \(p\) in \(M\) and point \(q\) in \(N\) respectively.
02

Define Tangient Space at the point in Product Manifold

Next, you need to define the tangent space at a point \((p,q)\) in the product manifold \(M \times N\). This is denoted by \(T_{(p,q)}(M \times N)\). It consists of all tangent vectors at point \((p,q)\) in \(M \times N\).
03

Define the Map

Then, define a map \(\Phi: T_{(p,q)}(M \times N) \rightarrow T_{p}(M) \oplus T_{q}(N)\) by letting \(\Phi(X_p, Y_q) = (X_p, Y_q)\), where \((X_p, Y_q) \) is an element in \(T_{(p,q)}(M \times N)\), \(X_p\in T_{p}(M)\) and \(Y_q \in T_{q}(N)\).
04

Showing the map is an isomorphism

We need to show that this map is indeed an isomorphism. That means it is linear, bijective and its inverse is linear as well. Linearity comes from the fact that the operation is a direct sum. To show that the map is bijective, we need to show that it is both injective and surjective. The function \(\Phi\) is injective if and only if \( (p,q) = (p',q') \) for \( (p',q')\in M \times N\). It is surjective because every element in \(T_{p}(M) \oplus T_{q}(N)\) comes from a unique element in \(T_{(p,q)}(M \times N) \).

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