Recurrent tensor

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In mathematics and physics, a recurrent tensor T with respect to a connection ∇ on a manifold M is one that satisfies the condition: there exists a one-form ω on M such that ∇T = ω ⊗ T. In other words, T changes only by a scaling dictated by ω.

Common examples
- Parallel tensors: If ∇T = 0, then ω = 0 and T is parallel. Parallel tensors are the simplest kind of recurrent tensors.
- Parallel vector fields: A vector field X with ∇X = 0 is recurrent. If X is recurrent and non-null with a closed one-form ω, X can be rescaled to a parallel vector field. Non-parallel recurrent vector fields are necessarily null.

Pseudo-Riemannian case
- On a pseudo-Riemannian manifold (M, g), the metric g is parallel with respect to the Levi-Civita connection (which has no torsion). Therefore, g is a recurrent tensor with ω = 0.

Weyl structures
- Weyl geometry studies connections that allow lengths to change during parallel transport but preserve angles up to a scale. A Weyl connection ∇′ satisfies ∇′g = φ ⊗ g for some one-form φ. In this setting, the metric g is recurrent with respect to ∇′.
- Under a conformal change g → e^λ g, the one-form transforms as φ → φ − dλ. This defines a canonical map from the conformal class [g] to the space of one-forms, linking conformal structure and Weyl geometry.

Other examples
- The curvature tensor R can also be recurrent in certain spacetimes, meaning ∇R = ω ⊗ R for some one-form ω.

In short, a recurrent tensor generalizes the idea of “unchanging up to scale” under parallel transport, capturing both fixed-quantity cases (like parallel tensors) and controlled, scale-based changes (as in Weyl structures).