The Riemann tensor is a multilinear operator of fourth rank acting on tangent vectors.
In metric theories of gravitation such as general relativity, curvature scalars play an important role in telling distinct spacetimes apart.
Two of the most basic curvature invariants in general relativity are the Kretschmann scalar and the Chern–Pontryagin scalar, These are analogous to two familiar quadratic invariants of the electromagnetic field tensor in classical electromagnetism.
An important unsolved problem in general relativity is to give a basis (and any syzygies) for the zero-th order invariants of the Riemann tensor.
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