Forthcoming Articles - Volume
Proceedings – Mathematical Sciences
PMSC-D-19-00428
Curvature, torsion and the quadrilateral gaps
Nitin Nitsure
For a manifold with an affine connection, we prove formulas which in-finitesimally quantify the gap in a certain naturally defined open geodesicquadrilateral associated to a pair of tangent vectors $u, v$ at a point of themanifold. We show that the 1st order infinitesimal obstruction to the quadri-lateral to close is always zero, the 2nd order infinitesimal obstruction to thequadrilateral to close is $-T(u, v)$ where $T$ is the torsion tensor of the con-nection, and if $T = 0$ then the 3rd order infinitesimal obstruction to thequadrilateral to close is $(1/2)R(u, v)(u + v)$ in terms of the curvature tensorof the connection. Consequently, the torsion of the connection, and if thetorsion is identically zero then also the curvature of the connection, can berecovered uniquely from knowing all the quadrilateral gaps. In particular, thisanswers a question of Rajaram Nityananda about the quadrilateral gaps ona curved Riemannian surface. The angles of $3\pi/4$ and $-\pi/4$ radians featureprominently in the answer, along with the value of the Gaussian curvature.This article is essentially self-contained, and written in an expository style.
Volume 130, 2020
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