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No: consider the vector fields $\frac{\partial}{\partial x}$ and $x\frac{\partial}{\partial y}$ on $\mathbb R^2$.

A finite covering map induces an injection on de Rham cohomology. Try searching for "integration along the fibers"; yours is an easy case, as you integrate on a finite set of points, which means you j …

Swan has proved that taking global section gives an anti-equivalence between finitely generate projective $\Gamma^{\infty}(M)$-modules and $C^{\infty}$ vector bundles on $M$; this correspondence is fu …

For 1): take a double covering $E\to B$, where $E$ and $B$ are compact oriented surface of genus 3 and 2 respectively, and give $E$ a structure of Riemann surface with trivial automorphism group. Abo …

"Data" is the plural form of the Latin word "datum", which means, among other things, "thing that is given". Viewed this way, it makes perfect sense, doesn't it?

The embedding of the unitary group $U_n$ into $GL_n(\mathbb C)$ is a homotopy equivalence; this is easily seen to imply that $H^*_{U_n}(X)$ is isomorphic to $H^*_{GL_n}(X)$. So the result for compact …