Curvature, torsion and the quadrilateral gaps

For a manifold with an affine connection, we prove formulas which infinitesimally quantify the gap in certain naturally defined open geodesic quadrilateral associated to pair of tangent vectors u, v at point manifold. We show that first-order infinitesimal obstruction close is always zero, second-order $$-T(u,v)$$ , where T torsion tensor and if $$T = 0$$ then third-order $$(1/2)R(u,v)(u+v)$$ terms curvature connection. Consequently, identically also connection can be recovered uniquely from knowing all gaps. In particular, this answers question Rajaram Nityananda about gaps on curved Riemannian surface. The angles $$3\pi /4$$ $$-\pi radians feature prominently answer, along value Gaussian curvature. This article essentially self-contained, written expository style.