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
(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(\esh \dashv \flat \dashv \sharp )$
dR-shape modality$\dashv$ dR-flat modality
$\esh_{dR} \dashv \flat_{dR}$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(\Re \dashv \Im \dashv \&)$
fermionic modality$\dashv$ bosonic modality $\dashv$ rheonomy modality
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
Models
Models for Smooth Infinitesimal Analysis
smooth algebra ($C^\infty$-ring)
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
Suppose $M$ is a smooth manifold. Recall that any differential $(k+1)$-form $K\in\Omega^{k+1}(M,T M)$ valued in the tangent bundle of $M$ gives rise to a graded derivation $\iota_K$ of degree $k$ on the algebra of differential forms on $M$: on 1-forms we have $\iota_K \omega=\omega\circ K$ and on higher forms we extend using the Leibniz identity.
Concretely,
where $Y_i=X_{\sigma(i)}$.
The map $\iota$ defined an injective homomorphism of graded vector spaces from $\Omega^{\bullet+1}(M,T M)$ to graded derivations of $\Omega(M)$. Its image comprises precisely those derivations that vanish on 0-forms and is closed under the commutator operation. Transferring the bracket to its source yields the Nijenhuis–Richardson bracket:
where $\iota_K(\omega\otimes X)=\iota_K \omega\otimes X$.
The Nijenhuis–Richardson bracket is an important ingredient in the classification of graded derivations of differential forms.
See the article Frölicher–Nijenhuis bracket for more information.
The formula for the NR bracket is made explicit in and thus derives its name from:
Albert Nijenhuis, Roger W. Richardson, Section 2 of: Cohomology and deformations of algebraic structures, Bulletin of the American Mathematical Society 70:3 (1964), 406–412. doi.
Albert Nijenhuis, Roger W. Richardson, Section 5 of: Deformations of Lie Algebra Structures, Journal of Mathematics and Mechanics 17 1 (1967) 89-105 [jstor:24902154]
but the existence of the bracket is already implicit in:
and refined to almost complex structures in
See also:
A textbook account: Chapter 16 of
Last revised on May 4, 2023 at 00:50:50. See the history of this page for a list of all contributions to it.