Let \(f: M \rightarrow N, g: N \rightarrow K\), and \(h=g: f: M \rightarrow K\). Show that \(h_{*}=g_{*} f_{*}\).

A pushforward is a linear map, that maps tangent vectors at one point to tangent vectors at another point using a given function. If \(f: M \rightarrow N\) is a smooth function between smooth manifolds \(M\) and \(N\), and \(p \in M\), then the pushforward of \(f\) at point \(p\) is denoted as \(f_{*p}: T_pM \rightarrow T_{f(p)}N\), where \(T_pM\) and \(T_{f(p)}N\) are the tangent spaces of \(M\) at point \(p\) and \(N\) at point \(f(p)\) respectively.

We will now apply the pushforward to the specific functions given in the exercise. 1. Since \(f: M \rightarrow N\), for a point \(p \in M\), we have \(f_{*p}: T_pM \rightarrow T_{f(p)}N\). 2. Similarly, for \(g: N \rightarrow K\), for a point \(q \in N\), we have \(g_{*q}: T_qN \rightarrow T_{g(q)}K\). However, in this case, we want to relate it to \(f\). Since \(p \in M\) and \(q \in N\), we can substitute \(q = f(p)\) and rewrite the pushforward of \(g\) at point \(f(p)\) as \(g_{*f(p)}: T_{f(p)}N \rightarrow T_{g(f(p))}K\). 3. Finally, for \(h = g \circ f: M \rightarrow K\), we have the pushforward of \(h\) at the point \(p\) as \(h_{*p} : T_pM \rightarrow T_{h(p)}K\). Since \(h = g \circ f\), we can rewrite this as \(h_{*p}: T_pM \rightarrow T_{g(f(p))}K\).

To show that \(h_{*} = g_{*} f_{*}\), we need to show that the pushforwards of \(h\), \(g\), and \(f\) are related in the way given by the exercise. Note that \(f_{*p} : T_pM \rightarrow T_{f(p)}N\) and \(g_{*f(p)}: T_{f(p)}N \rightarrow T_{g(f(p))}K\). They are properly set up for composition, and we can define their composition as \((g_{*f(p)} \circ f_{*p}): T_pM \rightarrow T_{g(f(p))}K\). From step 2, we have that \(h_{*p}: T_pM \rightarrow T_{g(f(p))}K\). We can now compare the definitions and see that the composition of pushforwards of \(g\) and \(f\) is the same as the pushforward of \(h\): \((g_{*f(p)} \circ f_{*p}) = h_{*p}\). Hence, we have shown that \(h_{*} = g_{*} f_{*}\).