Expression for Euler characteristic in differential geometry Is the Euler-Poincare$ ^{\prime}$ characteristic $\chi$ a bending invariant? If so, how should it be isometrically expressed in terms of first fundamental form coefficients?
Is not stretching invariant better as terminology? If so how can it be expressed in terms of the first and second form coefficients ?  and/or in any other way?
$\chi$

The Euler characteristic in a homotopy invariant, and that is a more precise  statement that the one you want.
The Gauss-Bonnet theorem states that the Euler characteristic of a closed orientable surface is equal, up to normalization, to the integral of the gaussian curvature.
More generally the Chern-Gauss-Bonnet theorem does the same thing for closed orientable manifolds of even dimension. Now you do not integrate the curvature but something derived from it. (The Euler characteristic of closed odd-dimensional manifoldsis zero, so nothing inteeresting can be said there)
This is very well explained in Spivak's book on differential geometry.
