A pullback metric is a way to define a Riemannian metric on a manifold by pulling back a metric from another manifold via a smooth map. This concept is crucial in understanding how geometric structures can be transferred between manifolds, particularly in the context of Riemannian submersions where one manifold projects onto another. The pullback metric allows us to study the properties of the original manifold through the lens of the structure on the target manifold.
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