Assume that at each point of $γ$ the vector field $f$ is either tangent or points toward the interior of $Ω$. Then $f$ has a zero inside $Ω$.

Question

Assume that at each point of $γ$ the vector field $f$ is either tangent or points toward the interior of $Ω$. Then $f$ has a zero inside $Ω$.

I know that If $f$ is tangent to the curve $\gamma$, then the index is 1. Then $f$ has a zero inside $\Omega$. How to prove another part of this theorem using index theory?

Answer 1

Note that with your new condition the index continues to be $1$.

Indeed, since the vector field cannot point inside, in case it moves from a "forward" tangent to a "bacward" tangent, it needs to go back to the original position since the curve closes. So, any amount that could contribute to a bigger/smaller index changes exactly in the same amount with the opposite sign.

Of course, you could use instead the Poincaré-Bendixson theorem.