Frölicher-Nijenhuis bracket

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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.

[edit] Definition

The Frölicher-Nijenhuis bracket is defined to be the unique vector-valued differential form

$[\cdot, \cdot] : \Omega^k(M,\mathrm{T}M) \times \Omega^l(M,\mathrm{T}M) \to \Omega^{k+l}(M,\mathrm{T}M) : (K, L) \mapsto [K, L]$

such that

$\mathcal{L}_{[K, L]} = [\mathcal{L}_K, \mathcal{L}_L]$

where $\mathcal{L}$ is the Nijenhuis-Lie derivative of $Ω(M)$ . The Frölicher-Nijenhuis bracket defines a graded Lie algebra structure on $Ω(M,T M)$ , which extends the Lie bracket of vector fields. Notably, Ω⁰(M, TM) consists of the space of sections of the tangent bundle (i.e., vector fields) on M. If K and L are both vector fields, then [K,L] agrees with the usual Lie bracket of vector fields.

[edit] Applications

The Nijenhuis tensor of an almost complex structure J, is the Frölicher-Nijenhuis bracket of J with itself. An almost complex structure is a complex structure if and only if the Nijenhuis tensor is zero.

With the Frölicher-Nijenhuis bracket it is possible to define the curvature and cocurvature of a vector-valued 1-form which is a projection. This generalizes the concept of the curvature of a connection.

The concept is named after mathematicians Alfred Frölicher and Albert Nijenhuis.

[edit] References

Frölicher, A. and Nijenhuis, A., Theory of vector valued differential forms. Part I., Indagationes Math 18 (1956) 338-360.
Frölicher, A. and Nijenhuis, A., Invariance of vector form operations under mapings, Comm. Math. Helv. 34 (1960), 227-248.