Riemannian 3-Manifolds with a Flatness Condition
The fundamental point-wise invariant of a Riemannian manifold $(M, g)$ is the Riemann curvature tensor. Many special types of Riemannian manifolds can be characterized by conditions on the Riemann curvature tensor and tensor fields derived from it. Examples include Einstein manifolds and conformally flat manifolds.
Here we restrict ourselves to three dimensions and explore the Remannian manifolds that arise when imposing conditions on the irreducible components of the first covariant derivative of the Riemann curvature tensor. Specifically, we look at an irreducible component of the covariant derivative which takes the form of a traceless symmetric $(0,3)$-tensor field. We classify the local and global structure of manifolds where this tensor field vanishes.
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