# Tag Info

### Existence of smooth functions decaying fast near the boundary

The following proof results from my discussion with Prof. John Lee. His comments and suggestions are very helpful. Step 1. First suppose that $U\subset\mathbb{R}^n$. Cover $U$ by countably many balls ...
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### Why specifying values on a local section is enough to determine the local values of a tensorial form on a principal $G$-bundle?

Staring at it again, it is not only possible to show that the monstrosity is vertical, it is in fact trivial to show that, just by applying $\pi_*$ to it and using $\pi \circ R_g = \pi$.
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### Parallel lines tangent to a strictly convex smooth plane body dividing its border in two arcs of equal length

Yes, it is true in general. Consider two arbitrary parallel lines $L1$ and $L2$ touching arbitrary strictly convex plane body $B$ with smooth border at points $P1$ and $P2$. Assume that arc $A1$, ...

### Flat metric defined by an abelian differential form.

In short, $f(z)$ has zeros(and poles), which make conic singularities, the metric is flat outside those singularities. Imagine a cube, which is flat except 8 vertices. You can search with quadratic ...
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### Is this Proof on 1-Form with Compact Support Correct?

Your proof is not correct in either the one dimensional case nor the higher dimensional case unfortunately. Here is what goes wrong in your proof (and alternative proofs) Your proof of the $1-$...
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### Understanding an integrable almost complex structure

An almost complex structure is simply a smooth choice of complex structure $J$ on each tangent bundle. This is a weaker statement than the manifold being a complex manifold itself since we haven't ...
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### How do I translate the description of geodesics of hyperbolic space in the hyperboloid model to the Poincaré ball and half-space models?

Here's how I would approach this: First, using the half-space and ball models only: Verify that in the half-space model vertical lines are geodesics and in the ball model lines through the origin are ...
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### Using parallel transport to construct geodesically convex regions

One natural way of defining Minkowski "addition" on a complete Riemannian manifold $M$ would, I think, be the following: Let $\Omega \subset M$ be a compact domain with Lipschitz boundary ...
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