Levi-Civita and Palatini showed that a unique symmetric linear connection can be recovered from a metric tensor alone via the Christoffel symbols without any auxiliary gauge field.

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Auxiliary(what you can revise to protect a belief from evidence): Supporting beliefs or assumptions that work alongside a main claim; if evidence contradicts your main idea, you can sometimes save it by changing these backup assumptions instead.
Christoffel symbols(differential geometry): Mathematical tools used to describe how space curves and bends, telling you how directions change as you move around a curved surface.
Gauge field(as used in differential geometry and physics): A mathematical tool that describes how forces or properties change smoothly across space, like how a compass needle's direction varies from place to place.
Levi-Civita(as a name referring to a specific mathematical technique): An Italian mathematician (Tullio Levi-Civita, 1873-1941) who developed mathematical tools for understanding how space curves and bends, which is essential to Einstein's theory of gravity.

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