• Curvature, torsion and the quadrilateral gaps

• # Fulltext

https://www.ias.ac.in/article/fulltext/pmsc/131/0004

• # Keywords

Affine connection; curvature; torsion; geodesics.

• # 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 first-order infinitesimal obstruction to the quadrilateral to close is always zero, the second-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 third-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.

• # Author Affiliations

1. School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400 005, India

• # Proceedings – Mathematical Sciences

Volume 131, 2021
All articles
Continuous Article Publishing mode

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019