I have not been able to do the following exercise, I have no idea where to start, can you help me please.
Let $ S(10) = \{e^{\frac{2\pi i k}{10}}: k = 0,1,..., 9 \} $. Prove or disprove that the topological subspace $ N_n (S (10)) = \{X \in C^{n \times n}: X^{\ast} X= XX^{\ast} \wedge \det(X) \in S(10) \} $ is path-connected with respect to the metric topology determined by the Frobenius metric on $ C^{n \times n}$.