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
(reduction modality infinitesimal shape modality infinitesimal flat modality)
\array{ && id &\dashv& id \ && \vee && \vee \ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \ && \bot && \bot \ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \ && \vee && \vee \ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \ && \bot && \bot \ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \ && \vee && \vee \ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \ && \bot && \bot \ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \ && \vee && \vee \ && \emptyset &\dashv& \ast }
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Models
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
∞-Lie theory (higher geometry)
The combination of an exterior derivative with a covariant derivative.
Let be a vector bundle with a linear connection given by the covariant derivative . We let be the space of all differential forms with values in? . To define the exterior covariant derivative, we take the explicit formula of the exterior derivative, and replace the usual derivative with the convariant derivative.
The exterior covariant derivative is defined by the following formula: given a -form , its exterior covariant derivative is given by
where each is a vector field on , and means omission of .
In the case of the trivial bundle with the trivial connection, this gives the usual exterior derivative of a vector-valued differential form.
Let be a principal -bundle. Let be a connection on . Let be the horizontal projection given by .
We let be a vector space, and be a representation of on .
A differential form is called:
horizontal, if whenever one of the vectors is vertical.
equivariant, if for all , where denotes the right action of on .
We denote by the space of horizontal and equivariant forms. Note that is in general not closed under the ordinary exterior derivative. There is a canonical isomorphism
where is the vector bundle associated to via the representation .
The exterior covariant derivative for forms on
is defined by
Every form in the image of is horizontal. If a form is equivariant, is also equivariant.
The restriction of to can be described in terms of the connection 1-form and the derivative of the representation :
Here we have used the following general notation: if are vector spaces, , and is a linear map, we have .
Unlike the usual exterior derivative, the exterior covariant derivative need not be nilpotent in general. Instead, we have
In particular, if is flat.
The exterior covariant derivative for forms on
is the map induced from under the isomorphism .
Note that a connection on the principal bundle induces? a connection on the associated vector bundle . Then the exterior covariant derivative in the sense of Definition coincides with this exterior covariant derivative.
The connection itself is not in : it is not horizontal.
The curvature? of is . Since , we have
The Bianchi identity is .
Under the isomorphism , the curvature corresponds to a 2-form , and the Bianchi identity corresponds to .
Last revised on December 1, 2017 at 19:11:27. See the history of this page for a list of all contributions to it.