There are works along the similar idea using vector fields.
I list two of them. There are more.
Reinhart, Bruce L.
The winding number on two-manifolds. (English) Zbl 0097.16203
Ann. Inst. Fourier 10, 271-283 (1960).
Chillingworth, D. R. J.
Winding numbers on surfaces. I. (English) Zbl...