A surface's Euler characteristic is related by the Gauss-Bonnet theorem to the integral of the Gaussian form of it over the surface.
The Cartan-Hadamard theorem says that the universal cover of a manifold where this is nonpositive is diffeomorphic to Euclidean space; more precisely, it requires that all the sectional ones are nonpositive. For a connection on a principal bundle, viewed as a gauge field, it is the field strength two-form. For a torus, it is intrinsically zero, but there can be an extrinsic one from the embedding in a Euclidean space. A surface's Euler characteristic is related by the Gauss-Bonnet theorem to the integral of the Gaussian form of it over the surface. The Riemann tensor is one measure of it. For the sphere, it is positive, while for a saddle, it is negative. For 10 points, what is this quantity, measuring an obstruction to mapping a space isometrically onto flat Euclidean space?
The Cartan-Hadamard theorem says that the universal cover of a manifold where this is nonpositive is diffeomorphic to Euclidean space; more precisely, it requires that all the sectional ones are nonpositive.
curvature (accept "sectional curvature" before the word "sectional")