0

This is a extension of my former question [Tangent bundle of open annulus is diffeomorphic to $\mathbb{S}^1 \times \mathbb{R}^3$

Now i want to show the tangent bundle $\mathbb{S}^1 \times \mathbb{R}^3$.

I tried to construct the similar method, but having trouble with manipulation of mobius strip.

Community
  • 1
phy_math
  • 6,448

1 Answers1

3

Let the mobius strip $M$ be defined as the quotient space $\{(\theta,y)|\theta\in[0,\pi],y\in \mathbb{R}\}/(0,y)\sim(\pi,-y).$ Then the tangent bundle of $M$, $TM$, can be characterized as $$TM=\{(\theta,y,A\frac{d}{d\theta},B\frac{d}{dy})|\theta\in[0,\pi],y,A,B\in \mathbb{R}\}/(0,y,A,B)\sim(\pi,-y,A,-B).$$ Thus we construct the map $TM\to S^1\times\mathbb{R}\times\mathbb{C}$ by $$(\theta,y,A,B)\mapsto (\theta,A,e^{i\theta}(y+\sqrt{-1}B)).$$ This map provides the desired diffeomorphism.

shrinklemma
  • 1,755