Non-vanishing vector field on S1xS2

[IAmAZucchini]

Homework Statement

Hi everyone,

This is a question on my tensor analysis/differential geometry homework due tomorrow, and I'm just not sure of the answer. The problem is to define a non-vanishing vector field V on S¹ x S².

Part b of the question is to "sketch a nonvanishing vector field on the Klein bottle".

Homework Equations

A vector field V : M $$\rightarrow$$ TM is nonvanishing if V(p) is never equal to the zero tangent vector in T_pM.

So if I'm not mistaken, S¹ x S² = { x$$\in$$R⁵ : x₁² + x₂² = 1, x₃² + x₄² + x₅² =1 }

The Attempt at a Solution

So I figure, if I combine nonvanishing vector fields on both S¹ andS², I'll get one for the S¹ x S² manifold. I know that on S¹, a nonvanishing vector field is just V(x₁,x₂) = (-x₂, x₁ ) because of the parametrization of a circle. However, a 2-sphere cannot be parametrized in terms of one coordinate, and even so, the hairy ball theorem states that it doesn't have a nonvanishing vector field. Will any vector field do, since the S¹ vector field I'm using is already nonvanishing? The textbook (The Geometry of Physics by Frankel) doesn't anything about nonvanishing fields. Even if this is the case, I can't think of the parametrization that'll give me a good tangent field, since it has to give 0 when dotted with (x¹,x²,x³)...

For part b) of the question (the Kelin bottle), I'm pretty sure that a constant vector field in the direction parallel to the sides identified in the same direction on the rectangle will work, since it's a smooth manifold. However, if someone could confirm that this is right, I'd really appreciate it...

Thanks for any help anyone offers!

_______
Alright, I had to hand it in, but if anyone knows the *right* answer, please do post it! I'll be going to office hours on thursday, probably, to find out anyways. My final answer was: since we can define a nonvanishing vector field on S1 V1 (x1, x2) -> (-x2,x1), and since there exists some vector field V2 tangent to S2, the combination of them V = (V1(x1, x2), V2(x3,x4,x5) ) will be nonvanishing since V1 is nonvanishing. Lame and vague/handwavy, but it's all I got.

 

[lippyka]

For part b, I said that a constant vector field in the direction of the edges identified will work, because it would be nonvanishing since it's smooth.