Curvature, torsion and the quadrilateral gaps

TL;DR

This paper derives formulas quantifying the infinitesimal gaps in geodesic quadrilaterals on manifolds with affine connections, revealing how torsion and curvature tensors can be uniquely recovered from these gaps.

Contribution

It provides explicit formulas linking quadrilateral gaps to torsion and curvature, answering a question about their geometric significance on curved surfaces.

Findings

First order gap is always zero.

Second order gap relates to torsion tensor T.

Third order gap relates to curvature tensor R.

Abstract

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