2 problems

Let $X$ be a $C^k$-manifold and $E$ a complex $C^k$ vector bundle of rank $r$ on $X$. Let $(y^1,\ldots,y^n)$ be a global coordinate system on $\Bbb R^n$, and let … be an assignment…

The $C^k$-map versus ring-homomorphism conjecture. The following statements are equivalent: (1) $\varphi^{\sharp}$ is a ring-homomorphism over ${\Bbb R}\hookrightarrow{\Bbb C}$; (2…