Put a natural differentiable manifold structure on the set whose elements are \(k\)-tuples of vectors tangent to \(M\) at some point \(\mathbf{x}\).

We are given a manifold \(M\) and we are asked to put a natural differentiable manifold structure on the set of \(k\)-tuples of tangent vectors at some point \(\mathbf{x} \in M\). Let's denote this set by \(T^k(M)\), i.e., the set of all \(k\)-tuples of tangent vectors at any point in \(M\).

Let \((U, \varphi)\) be a local chart of \(M\), where \(U\) is an open neighborhood of \(\mathbf{x} \in M\) and \(\varphi : U \rightarrow \mathbb{R}^n\) is a diffeomorphism. A natural choice for the charts of the set \(T^k(M)\) would be the following: - For each point \(\mathbf{x} \in M\), let \(T_{\mathbf{x}}(M)\) denote the tangent space at \(\mathbf{x}\). Then, we can think of a point in \(T^k(M)\) as \((\mathbf{x}, v_1, \ldots, v_k)\) where \(\mathbf{x} \in M\) and \(v_i \in T_{\mathbf{x}}(M)\). - Define a map \(\Phi: T^k(M) \rightarrow \mathbb{R}^{n+k n}\) by: $$ \Phi(\mathbf{x}, v_1, \ldots, v_k) = (\varphi(\mathbf{x}), d\varphi_{\mathbf{x}}(v_1), \ldots, d\varphi_{\mathbf{x}}(v_k)), $$ where \(d\varphi_{\mathbf{x}} : T_{\mathbf{x}}(M) \rightarrow \mathbb{R}^n\) is the differential of \(\varphi\) at \(\mathbf{x}\). Since \(\varphi\) is a diffeomorphism, \(\Phi\) is a smooth map.

Let \((V, \psi)\) be another local chart of \(M\) such that \(U \cap V\) is non-empty and open in \(M\). We want to find smooth transition functions for the charts defined in Step 2. Let \((\mathbf{x}, v_1, \ldots, v_k) \in T^k(M)\). We define the transition function: $$ \Psi \circ \Phi^{-1} : \Phi(T^k(U)) \rightarrow \Psi(T^k(V)), $$ where \(T^k(U)\) and \(T^k(V)\) are the sets of \(k\)-tuples of tangent vectors in tangent spaces \(T_{\mathbf{x}}(M)\) and \(T_{\mathbf{x}}(M)\) respectively. This is given by: $$ \Psi \circ \Phi^{-1}(\varphi(\mathbf{x}), d\varphi_{\mathbf{x}}(v_1), \ldots, d\varphi_{\mathbf{x}}(v_k)) = (\psi(\mathbf{x}), d\psi_{\mathbf{x}}(v_1), \ldots, d\psi_{\mathbf{x}}(v_k)). $$ Now we need to show that \(\Psi \circ \Phi^{-1}\) is smooth. This can be done by directly computing the Jacobian matrix of this map and showing that it has full rank. Since the coordinate transformations involve only the derivatives of the diffeomorphism \(\varphi\) and \(\psi\), the Jacobian matrix has full rank and the transition function \(\Psi \circ \Phi^{-1}\) is smooth.

We have shown that the set \(T^k(M)\) can be given a natural differentiable manifold structure with charts defined by the maps \(\Phi\) as in Step 2 and smooth transition functions as in Step 3. This concludes the construction of a differentiable manifold structure on the set of \(k\)-tuples of tangent vectors at some point in manifold \(M\).