The Gauss-Bonnet theorem is usually seen without the geodesic curvature $k_g$ line integral, for compact manifolds M without boundary (making the Euler characteristic equal the area integral of the Gaussian curvature $K$). That is: $$ 2\pi\chi(M)=\iint_M K dA + \int_{\partial M} k_g dS = \iint_M K dA + 0. $$
Let's say that I have an arbitrary 2D manifold X without boundary (as in the usual case above). Inside it, I choose a counter-clockwise-oriented closed loop (so that the tangential direction is chosen per convention), and use the Gauss-Bonnet (GB) theorem on the manifold Y dictated by the loop. Essentially, this is like choosing from the main manifold X, a submanifold Y.
How does the Euler characteristic from using the GB theorem on submanifold Y $\chi(Y)$ relate to that of the parent manifold X $\chi(X)$? In this situation, what does the resulting Euler-characteristic mean? When will $\chi(X) = \chi(Y)$, if at all? Or, are there any other general relationships? I guess my confusion is whether the GB theorem describes the same manifold or two different ones. For example, does the geodesic curvature integral of Y compensate for the missing area for the gaussian curvature integral, making both Euler characteristics equal?
The most I have found in the literature has been about two disjoint manifolds bounded by curves summing to equal the overall Euler character of the system.