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)
For $X$ a smooth manifold and $T X$ its tangent bundle a multivector field on $X$ is an element of the exterior algebra bundle $\wedge^\bullet_{C^\infty(X)}(\Gamma(T X))$ of skew-symmetric tensor powers of sections of $T X$.
In degree $0$ these are simply the smooth functions on $X$.
In degree $1$ these are simply the tangent vector fields on $X$.
In degree $p$ these are sometimes called the $p$-vector fields on $X$.
In suitable contexts, multivector fields on $X$ can be identified with the Hochschild cohomology $HH^\bullet(C(X), C(X))$ of the algebra of functions on $X$.
There is a canonical bilinear pairing on multivector fields called the Schouten bracket.
Let $X$ be a smooth manifold of dimension $d$, which is equipped with an orientation exhibited by a differential form $\omega \in \Omega^d(X)$.
Then contraction with $\omega$ induces for all $0 \leq n \leq d$ an isomorphism of vector spaces
The transport of the de Rham differential along these isomorphism equips $T^\bullet_{poly}$ with the structure of a chain complex
The operation $div_\omega$ is a derivation of the Schouten bracket and makes multivectorfields into a BV-algebra.
A more general discussion of this phenomenon in (Cattaneo–Fiorenza–Longoni). Even more generally, see Poincaré duality for Hochschild cohomology.
Just as for vector fields one has the notion of an integral curve, so for $k$-vector fields one has a generalization known as integral sections. Given a $k$-vector field $X$ on a manifold $M$, and a point $p\in M$, an integral section is a map $\phi: U\subset \mathbb{R}^k \to M$ such that $\phi(0)=p$ and
See e.g. Section 3.1 in de León et al. 2015 and the references cited therein for more.
The isomorphisms between the de Rham complex and the complex of polyvector field is reviewed for instance on p. 3 of
and in section 2 of
and on p. 6 of
An exposition of multivector fields and their use in Lagrangian and Hamiltonian theories:
Last revised on April 18, 2024 at 16:44:53. See the history of this page for a list of all contributions to it.