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The simple fact in question is false in any dimension greater than one. Consider the strip $ \mathbb{R} \times [-\pi/2,\pi/2] \subset \mathbb{R}^2$. At a point $(x, y)$ take the vector $(-sin(y), co …

Let $C$ be a complex curve. Recall that a Higgs bundle on $C$ is a vector bundle $E$ on $C$ equipped with a morphism $E \to E \otimes K_C$. The space of (stable) Higgs bundles is much studied, and i …

Say I stand at the north pole and talk; in sufficiently frictionless conditions, one imagines that someone standing at the south pole could listen. More generally, say $M$ is a compact Riemannian ma …

A comment re. Jonny's nice answer: there was indeed a time when that was the envisioned strategy of proof. However our present approach does not require the arborealization. Because: now we know tha …

In geometric measure theory, there’s an answer to the question “what’s the limit of a family of submanifolds”, namely there’s some kind of object called an integral current. In the geometric question …

In mathematics, sheaves can often be resolved by vector bundles. That is, given a sheaf $\mathcal{F}$, one can find vector bundles $E_i$ and an exact sequence: $$\cdots \to E_2 \to E_1 \to E_0 \to \ …