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Differential geometry, which in simple terms is the geometry of curvature, has been of increasing importance to mathematical physics since the suggestion that space is not flat space. Contemporary differential geometry is intrinsic, meaning that space is a manifold and structure is given by a Riemannian metric, or analogue, locally determining a geometry that is variable from point to point.
This approach contrasts with the extrinsic point of view, where curvature means the way a space bends within a larger space. The idea of 'larger' spaces is discarded, and instead manifolds carry vector bundles. Fundamental to this approach is the connection between curvature and characteristic classes, as exemplified by the generalized Gauss-Bonnet theorem.
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