# Tagged Questions

For questions on manifolds of dimension $n$, a topological space that near each point resembles $n$-dimensional Euclidean space.

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### Obtaining embedding from geodesic

Suppose $M$ is an embedded sub-manifold of $D$-dimensional Euclidean space $E^D$, with embedding $\phi:M \hookrightarrow E^D$. And suppose I know the induced Riemmanian-metric $g$ on $M$, which ...
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### The relationship between dimension of a manifold and coordinate function

I am thinking about the intrinsic meaning (what this equation really means) about this equation. Suppose $\mathcal M$ is a smooth manifold embedded in $\mathcal R^d$, then for any $x \in \mathcal M$,...
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### Existence of a universal cover of a manifold.

Suppose $M$ is a manifold, topological or smooth etc. As a topological space $M$ is required to be primarily locally homeomorphic to $\Bbb R^n$, with some added things that don't come along with this, ...
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### Need help understanding part of this proof about local coordinates for Legendrian manifold

I need help understanding this proof in this book here: Concretely, I do not understand why it is okay to assume that $S$ can be parameterized by $n$ variables. Sure, it's an $n$-dimensional ...
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### Geodesics on $SO(n)$

I'm trying to prove the following exercise about the ortogonal symmetric group $SO(n)$. I have been able to prove the first two sections of the exercise but I got stuck on the third. I don't ...
Suppose we have a smooth map$f:SO(3) → SO(3)$ of manifolds s.t.$f(X)=X^2$. $I$ though since I is a regular value of this map and f is orientation preserving, to calculate degree of it, it is enough ...