∞-Lie theory (higher geometry)
Background
Smooth structure
Higher groupoids
Lie theory
∞-Lie groupoids
∞-Lie algebroids
Formal Lie groupoids
Cohomology
Homotopy
Related topics
Examples
-Lie groupoids
-Lie groups
-Lie algebroids
-Lie algebras
The notion of Ehresmann connection is one of the various equivalent definitions of connection on a bundle.
Let be a Lie group with Lie algebra and a -principal bundle. Let
be the action of on and
its derivative, sending each element to the vector field on that at is the push-forward .
For and a differential form on write for the contraction.
Given a -principal bundle over , a Cartan-Ehresmann connection on is a Lie algebra-valued 1-form
on the total space satisfying two conditions:
first Ehresmann condition
for every we have
second Ehresmann condition
for every we have
where is the Lie derivative along and where is the adjoint action of on itself.
This is equivalent to
first Ehresmann condition
for every we have
second Ehresmann condition
for every we have
where is the curvature 2-form of .
Using we have by Cartan calculus
Given a smooth bundle with typical fiber (e.g. a smooth vector bundle or a smooth principal -bundle), there is a well defined vector subbundle over such that consists of the tangent vectors such that . A smooth distribution (field) of horizontal subspaces is a choice of a vector subspace for every such that
E1. (complementarity)
E2. is smooth. That means that in the unique decomposition of any smooth vector field on into vector fields and such that the vector field is smooth (or equivalently is smooth, or equivalently both) as a section of (there exist yet several other equivalent formulations of the smooothness criterion).
An Ehresmann connection describes a connection on a -principal bundle (for some Lie group) in terms of a distribution of horizontal subspaces which is a subbundle of the tangent bundle of complementary at each point to the vertical tangent bundle to the fiber. More precisely, an Ehresmann connection on a principal -bundle is a smooth distribution of horizontal subspaces which is equivariant:
E3. for every and .
This subbundle over can be expressed as a field of subspaces () which are pointwise annihilators of a smooth Lie algebra-valued -form on that satisfies two Ehresmann conditions from the previous subsection.
The Ehresmann connections on a principal -bundle are in 1-1 correspondence with an appropriate notion of a connection on the associated bundle. Namely, if is the smooth horizontal distrubution of subspaces defining the principal connection on a principal -bundle over , where is a Lie group and a smooth left -space, then consider the total space of the associated bundle with typical fiber . Then, for a fixed one defines a map assigning the class to . If defines the horizontal subspace , the collection of such subspaces does not depend on the choice of in the class , and the correspondence is a connection on the associated bundle .
One may also describe(flat) Ehresmann connections in cohesive homotopy type theory.
The general abstract discussion is here. The discussion of how in smooth infinity-groupoids this reduces to the traditional notion is here.
The two definitions in terms of 1-forms and in terms of horizontal distributions are equivalent.
At each take the horizontal subspace to be the kernel of
This means we may think of as measuring how infinitesimal paths in fail to be horizontal or parallel to in the sense of parallel transport.
Let be an invariant polynomial on the Lie algebra. For an Ehresmann connection, write
for the curvature characteristic form obtained by evaluating this on wedge powers of the curvature 2-form.
The forms are closed, descend along , in that they are pullbacks of forms along , and their class in de Rham cohomology are independent of the choice of on .
That the forms are closed follows from the Bianchi identity
satisfied by the curvature 2-form and the defining as-invariance of . More abstractly, the 1-form itself may be identified with a morphism of dg-algebras out of the Weil algebra (see there)
and the evaluation of the curvature in the invariant polynomials corresponds to the precomposition with the morphism
described at ∞-Lie algebra cohomology.
to show that these forms descend, it is sufficient to show that for all we have
The first follows from . The second from this, the -closure just discussed and Cartan's magic formula for the Lie derivative.
The form is called the curvature characteristic form of the connection . The map
induced by as above is the Chern-Weil homomorphism.
The terminology for the various incarnations of the single notion of connection on a bundle varies throughout the literature. What we here call an Ehresmann connection is sometimes, but not always, called principal connection (as it is defined for principal bundles).
The notion originates with:
Charles Ehresmann, Les connexions infinitésimales dans un espace fibré différentiable,
Colloque de topologie (espaces fibrés), Bruxelles, 1950, Georges Thone, Liège; Masson et Cie., Paris (1951) 29–55
Séminaire Bourbaki, 1 24 (1952) 153–168 [numdam:SB_1948-1951__1__153_0, pdf]
See also:
A formulation and discussion of Ehresmann connections using language and tools from synthetic differential geometry:
Generalization to principal 2-bundles:
Generalization to connections on principal ∞-bundles:
A more comprehensive account is in sections 3.9.6, 3.9.7 of
Last revised on February 1, 2024 at 10:47:56. See the history of this page for a list of all contributions to it.