  In four dimensions one can use the chiral part of the spin connection as the
main object that encodes geometry. The metric is then recovered algebraically
from the curvature of this connection. We address the question of how
isometries can be identified in this "pure connection" formalism. We show that
isometries are recovered from gauge transformation parameters satisfying the
requirement that the Lie derivative of the connection along a vector field
generating an isometry is a gauge transformation. This requirement can be
rewritten as a first order differential equation involving the gauge
transformation parameter only. Once a gauge transformation satisfying this
equation is found, the isometry generating vector field is recovered
algebraically. We work out examples of the new formalism being used to
determine isometries, and also prove a general statement: a negative definite
connection on a compact manifold does not have symmetries. This is the precise
"pure connection" analog of the well-known Riemannian geometry statement that
there are no Killing vector fields on compact manifolds with negative Ricci
curvature.
