# Tagged Questions

For questions regarding the structure and properties of compact manifolds.

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### How can I show $R^n$ is dense in $S^n$?

How can I show $R^n$ is dense in $S^n$? I wanted to show $S^n$ is compactification of $R^n$. for this I need $R^n$ is not compact, for this there is no problem, and $S^n$ is compact, I did it with ...
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### Momentum is quantised in compact spaces?

Background One of the first examples given when studying quantum mechanics is the particle on a cylinder, or particle on a ring. One finds that because of the periodic boundary conditions, ...
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### Proof that a map from an orientable surface to a non-orientable surface has even degree.

For a smooth map $f:M\to N$ from an orientable closed surface $M$ to a non-orientable closed surface $N$, we define its parity (also called modulo 2 degree, and denoted $\deg_2(f)$) as the parity of ...
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### How to qualify a N dimensional manifold as Compact under following condions?

Suppose a manifold of N dimensions is closed and bounded in a dimension but it remained unbounded in all other dimensions, so how to categorize the manifold. For example, in simpler form how to ...
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### Partition of unity. Does this one exist?

Let $X:=\mathbb{R^n}$ be given and $M \subset X$ be a compact set in it. Then my question is: Are there $\alpha_i \in C^{\infty}(X,\mathbb{R})$ such that $supp(\alpha_i) \subset N$, where $N$ is an ...
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### When are flag manifolds compact?

This is a question from Lee's book on Smooth Manifolds, question 21-16: Let $F_K(V)$ be the set of flags of type $K$ in a finite-dimensional (real) vector space $V$. Show that $GL(V)$ acts ...
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