The curvature operator is a mathematical object that encodes the intrinsic curvature of a Riemannian manifold by mapping pairs of tangent vectors to a new tangent vector, essentially measuring how the geometry of the manifold deviates from being flat. This operator plays a critical role in understanding the geometric properties of manifolds, particularly through local coordinates, as it can be expressed in terms of the Riemann curvature tensor, leading to various applications and examples that illustrate curvature behavior in different scenarios.
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