An induced metric is a way to define a Riemannian metric on a submanifold by pulling back the metric from the ambient space where the submanifold is embedded. It captures the intrinsic geometric properties of the submanifold, allowing one to measure distances and angles solely based on the structure of the submanifold itself, rather than the larger space it resides in. This concept highlights the relationship between a manifold and its surrounding environment.
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