A constant motion

This is a mathematical physics question, which is a little different from our usual fare. We suspect some of you will need to do a little research on this and so we hope you learn a bit.

You live in two spatial dimensions, and your world is a smooth, closed two-dimensional surface (instead of our usual world with 3 spatial dimensions). A closed two-dimensional surface could be a sphere, torus, torus with more than one hole, Klein bottle, etc. In your world, you possess something unique - you are the only person to possess an electric charge. Unfortunately, your charge is very, very small in magnitude: it is a test charge with magnitude \(dq\). You'd like to put your test charge down and take a nap, but you notice that no matter where you place your test charge (anywhere on the surface), it will accelerate (albeit with a small acceleration) away from you in some direction.

What is the Euler characteristic of the surface you live on?

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