# Tagged Questions

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### Is T($S^2 \times S^1$) trivial?

How would I find out if T($S^2 \times S^1$) is trivial or not? Using the hairy ball theorem I can show that T($S^2$) is not trivial, and it is straight forward to show that T($S^1$) is trivial. ...
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### Definition of a coordinate vector bundle

Consider the following definition of a coordinate vector bundle. Let $M$ be a smooth manifold of dimension $m$, and $\{(f, U_f)\}$ an atlas of compatible charts for $M$. A smooth coordinate ...
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### What would be the space of section of the bundle $\mathfrak{g}\longrightarrow \{e\}$?

Let $\mathfrak{g}$ be a Lie algebra and $\pi:\mathfrak{g}\longrightarrow \{e\}$a vector bundle over a point. What would be the sections of this bundle?
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### a question about compact tangent bundle

I have a question about tangent bundles. Is there a compact tangent bundle? Or what conditions do we need to be sure that tangent bundle of a manifold be compact?
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### Tangent bundle of sphere with $g$ handles

How can one show that tangent bundle $TM$ is not trivial if $M$ is a sphere with $g$ handles and $g \ne 1$?
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### How to prove that a vector bundle is trivial iff there are n global sections that form a basis on each fiber?

I can prove the only if part. My attempt to prove if part is the following: Given $n$ global sections $s_1, s_2, ..., s_n$ of a vector bundle $E$ on a smooth manifold $M$ such that they form a basis ...
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### Cotangent bundle of n-dimensional diferentiable manifold is 2n-dimensional manifold

How to prove that cotangent bundle of n-dimensional diferentiable manifold is 2n-dimensional manifold? Detailed explanation is welcome. Thanks in advance.
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### Constructing vector bundles from local covers and transitions functions

Let $M$ be a smooth manifold. Suppose we are given an open cover ${U_\alpha}$ of $M$ and for $\alpha,\beta$ ; a smooth map $\tau_{\alpha\beta}\colon U_\alpha \cap U_\beta \to GL(k; R)$ satisfying the ...
### A good way to embed a manifold in a Euclidian space $\mathbb{R}^n$
We know that any closed manifold $X$ can be embedded into some Euclidian space $\mathbb{R}^n$ for sufficiently large $n\in \mathbb{N}$. What is the easiest way to see this fact? I have seen several ...