One can define \(k\)-forms in a slightly different way by associating with each tangent space \(T_{m} M\) the vector space \(\bigwedge^{k} T_{m}^{*} M\). The elements of \(\bigwedge^{k} T_{m}^{*} M\) are maps $$ \alpha: \overbrace{T_{m} M \times \cdots \times T_{m} M}^{k} \rightarrow \mathbb{R} $$ with the properties (a) they linear over \(\mathbb{R}\) in each argument. (b) they change sign when any two arguments are interchanged.

First, let's briefly discuss the following objects involved in the problem: - Tangent space, \(T_{m}M\), is a vector space containing all tangent vectors at a point \(m\) on the manifold \(M\). - A covector, also known as a one-form, maps a vector to a real number. - \(k\)-form, represented by \(\bigwedge^{k} T_{m}^{*}M\), is a map that takes \(k\) tangent vectors and returns a real number. It has the properties (a) and (b) as stated in the exercise. The goal of the exercise is to analyze and prove these two properties of \(k\)-forms.

To demonstrate property (a), we must show that k-forms are linear over \(\mathbb{R}\) in each argument, meaning they obey the addition and scalar multiplication properties. Suppose \(\alpha\) is a \(k\)-form in \(\bigwedge^{k} T_{m}^{*} M\), and let \(\mathbf{v_1,\dots, v_k}\) be tangent vectors in \(T_{m} M\). For simplicity, we will be proving linearity for the first argument, but the same argument can be used for every position. Consider two tangent vectors, \(\mathbf{u_1}\) and \(\mathbf{u_2}\), in \(T_{m} M\), and a scalar \(c \in \mathbb{R}\). We want to show that $$ \alpha(\mathbf{u_1} + \mathbf{u_2}, \mathbf{v_2}, \dots, \mathbf{v_k}) = \alpha(\mathbf{u_1}, \mathbf{v_2}, \dots, \mathbf{v_k}) + \alpha(\mathbf{u_2}, \mathbf{v_2}, \dots, \mathbf{v_k}) $$ and $$ \alpha(c \mathbf{u_1}, \mathbf{v_2}, \dots, \mathbf{v_k}) = c \alpha(\mathbf{u_1}, \mathbf{v_2}, \dots, \mathbf{v_k}). $$ If \(\alpha\) satisfies both of these conditions for the first argument, then it is linear over \(\mathbb{R}\) in that argument. The same can be done for each argument in a similar manner, proving property (a).

To prove property (b), we must show that the k-form changes sign when any two arguments are interchanged. We can demonstrate this by considering two adjacent arguments. Let \(\alpha\) be a \(k\)-form in \(\bigwedge^{k} T_{m}^{*} M\), and let \(i, j \in \{1,\dots,k\}\) with \(i <j\). Consider two tangent vectors \(\mathbf{v_i}\) and \(\mathbf{v_j}\) in \(T_{m} M\). We want to show that $$ \alpha(\dots, \mathbf{v_i}, \dots, \mathbf{v_j}, \dots) = -\alpha(\dots, \mathbf{v_j}, \dots, \mathbf{v_i}, \dots). $$ If \(\alpha\) satisfies this condition, it means that the sign changes whenever two arguments are interchanged, thus proving property (b). In conclusion, we defined \(k\)-forms as maps that take \(k\) tangent vectors and return a real number, and we demonstrated their basic properties: linearity over \(\mathbb{R}\) in each argument and changing signs when any two arguments are interchanged.