Non-vanishing vector fields on the 2 Torus

Question

Can you explicitly construct several non vanishing vector fields on the 2 Torus $S^1 \times S^1$? How can you build non zero vector fields in this particular case? Thank you in advance.

Answer 1

In general, on any manifold, given any two independent vector fields, you can take linear combinations of them to get lots of others.

So, take the vector field $\frac{d}{d\theta}$ pointing along the first circle, and the vector field $\frac{d}{d\phi}$ pointing along the second circle. Now form linear combinations $r \cdot \frac{d}{d\theta} + s \cdot \frac{d}{d\phi}$ to get infinitely many nowhere zero vector fields.

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Added: Illustrated examples:

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Answer 2

Hint: Think about one of the two canonical circles in the torus and how you can foliate a the torus with it.