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.

中文翻译：

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来自规范变换的等距线

在四个维度中，可以使用自旋连接的手征部分作为编码几何的主要对象。然后从这个连接的曲率中以代数方式恢复度量。我们解决了如何在这种“纯连接”形式主义中识别等距的问题。我们表明，等距是从规范变换参数中恢复的，该参数满足以下要求，即沿着生成等距的矢量场的连接的李导数是规范变换。该要求可以重写为仅涉及规范变换参数的一阶微分方程。一旦找到满足该方程的规范变换，等距生成矢量场就可以代数恢复。我们制定了用于确定等距的新形式主义的例子，并证明一个一般命题：紧流形上的负定连接不具有对称性。这是众所周知的黎曼几何陈述的精确“纯连接”模拟，即在具有负 Ricci 曲率的紧凑流形上没有杀伤向量场。