Frölicher–Nijenhuis bracket

In mathematics, the Frölicher–Nijenhuis bracket is an extension of the Lie bracket of vector fields to vector-valued differential forms on a differentiable manifold.

It is useful in the study of connections, notably the Ehresmann connection, as well as in the more general study of projections in the tangent bundle. It was introduced by Alfred Frölicher and Albert Nijenhuis (1956) and is related to the work of Schouten (1940).

It is related to but not the same as the Nijenhuis–Richardson bracket and the Schouten–Nijenhuis bracket.

Let Ω*(M) be the sheaf of exterior algebras of differential forms on a smooth manifold M. This is a graded algebra in which forms are graded by degree:

A graded derivation of degree ℓ is a mapping

which is linear with respect to constants and satisfies

Thus, in particular, the interior product with a vector defines a graded derivation of degree ℓ = −1, whereas the exterior derivative is a graded derivation of degree ℓ = 1.

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