11
votes
2answers
378 views

Vector field on 3-sphere

Let $V$ be a vector field on $S^3$ such that its singularity points, namely the points at which the vector field vanishes, are only sinks or sources (i.e. the field is converging or diverging). Is ...
3
votes
2answers
211 views

Unstable Foliations

Let $M$ be a closed compact Riemannian manifold, $\mathcal{F}$ be a $C^1$ foliation on $M$. Let $F(x)\in\mathcal{F}$ be the leaf containing $x$. Definition. $\mathcal{F}$ is said to be a unstable ...
4
votes
1answer
183 views

Mapping torus relative to an infinite orbit can be hyperbolic with finite volume?

Consider a homeomorphism of the sphere with an infinite orbit converging forwards and backwards to the same point. Remove the orbit and the accumulation point and make the mapping torus. Can the ...
4
votes
1answer
277 views

what is the meaning of “inseparable” in this case

Let $i:T^{2}\rightarrow S^{1}\times S^{2}$ be an embedding map. If $i(T^{2})$ is inseparable in $S^{1}\times S^{2}$, then $S^{1}\times S^{2}-i(T^{2})\cong (O\cup K)^{c}$. Here $(O\cup K)^{c}$ is a two ...