Try to talk about orientation in free geometry. Also, try finding eigenvalues for the eigenspaces of the Laplacian on [math]S^{N-1}_{\mathbb R,+}[/math]. Also, try constructing a spectral triple in the sense of Connes for [math]S^{N-1}_{\mathbb R,+}[/math], and explain what works, and what fails. And finally, try proving a Nash theorem for the spectral triples in the sense of Connes.