Weitzenbock's absolute parallelism and teleparallel gravity theories exploit precisely this extra structure to define connections inequivalent to the Levi-Civita connection despite sharing the same metric.

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Absolute parallelism(as a geometric framework in physics): A geometric theory where lines or directions remain parallel to each other throughout space, rather than converging or diverging as they might in curved space.
Connection (in mathematics)(as a geometric structure): A mathematical tool that describes how to compare directions and measurements at different points in space—think of it as a rule for how geometry works.
Inequivalent(describing different connections with the same metric): Not the same or not interchangeable; two things that look different even if they share some properties.
Levi-Civita connection: The uniquely compatible linear symmetric connection associated with the metric tensor
Metric(in mathematical analysis): A specific way of measuring distance or defining what 'closeness' means in a mathematical space.
Teleparallel gravity(as an alternative gravity theory): An alternative theory of gravity (like Einstein's relativity) that explains how gravity works using the idea of absolute parallelism instead of curved spacetime.
Weitzenbock (Weitzenbäck)(as the namesake of absolute parallelism theory): A German mathematician who developed theories about geometry and gravity; here his name is used to describe specific mathematical approaches to understanding how space and forces work.

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