Frölicher-Nijenhuis bracket

Encyclopedia : F : FR : FRL : 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.

Definition

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

[[cdot, cdot] : \Omega^k(M,\mathrmM) \times \Omega^l(M,\mathrmM) \to \Omega^(M,\mathrmM) : (K, L) \mapsto [K, L]]

such that

[\mathcal_ = [mathcal_K, mathcal_L]]

where [\mathcal] is the Nijenhuis-Lie derivative of [\Omega(M)]. The Frölicher-Nijenhuis bracket defines a graded Lie algebra structure on [\Omega(M,\mathrmM)], 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.

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.

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.

From Wikipedia, the Free Encyclopedia. Original article here. Support Wikipedia by contributing or donating.
All text is available under the terms of the GNU Free Documentation License See Wikipedia Copyrights for details.