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3

I agree with Mike that you should ask a faculty member about this, since we have no concrete idea about how deep your lecture program goes, nor what is of interest to your professor. That being said, there is an answer lying perfectly in the intersection of your two areas, and that is Klein Geometry. Klein started the so called Erlangen Program in the ...

2

Yes, if $f$ is not injective, there are always infinitely many such pairs - if the dimension of $M$ is $> 0$, for $\dim M = 0$ there are only finitely many such pairs since a compact manifold of dimension $0$ is finite. Let $a,b \in M$ with $f(a) = f(b)$. If $f$ is not injective on any neighbourhood of $a$, then we find infinitely many pairs accumulating ...

1

Let $C$ be a component of $q^{-1}(A)$ and let $p:C\to A$ be the restriction of $q$. Now if $x\in A$, intersect the evenly covered neighborhood of $x$ with $A$, and choose a smaller path connected neighborhood $U$ of $x$ relative to $A$. Then $U$ is evenly covered. The path components of $q^{-1}(U)$ are mapped homeomorphically onto $U$. Since $C$ is a path ...

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