synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
Differentials
Tangency
The magic algebraic facts
Theorems
Axiomatics
Models
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
Suppose is a smooth manifold. Recall (e.g. from the article Nijenhuis–Richardson bracket) that any differential -form valued in the tangent bundle of gives rise to a graded derivation of degree on the de Rham algebra of differential forms on : on 1-forms we have and on higher forms we extend using the Leibniz rule.
Concretely,
where the sum is over all permutations and where denotes the sign of the permutation.
makes it natural to define the Lie derivative with respect to :
The map defines an injective homomorphism of graded vector spaces from to graded derivations of . Its image comprises precisely those derivations such that and is closed under the commutator operation. Transferring the bracket to its source yields the Frölicher–Nijenhuis bracket:
for a uniquely defined .
Taken together, the Frölicher–Nijenhuis bracket and the Nijenhuis–Richardson bracket allows us to fully classify the graded derivations of the de Rham algebra of differential forms on :
A graded dervation of degree on has a unique presentation of the form
where , .
We have if and only if and if and only if vanishes on 0-forms.
Finally, the graded commutator of graded derivations can be expressed in terms of the Frölicher–Nijenhuis bracket (for ) and the Nijenhuis–Richardson bracket (for ):
Given a smooth manifold and differential forms , valued in the tangent bundle of , their Frölicher–Nijenhuis bracket is a differential form
defined by the formula
and is the sign of the permutation .
The Nijenhuis tensor of an almost complex structure is . The explicit formula yields
The original definition, with an explicit formula is in Section 6 of
Further development:
Refinements for almost complex structures:
Discussion as a natural operation:
An article in Encyclopedia of Mathematics:
A textbook account: Chapter 16 of
Last revised on May 4, 2023 at 00:50:10. See the history of this page for a list of all contributions to it.