Horizontal bundle
Encyclopedia
In mathematics
, in the field of differential topology
, given
a smooth fiber bundle
over a smooth manifold M, then the vertical bundle
VE of E is the subbundle of the tangent bundle
TE consisting of the vectors which are tangent to the fibers of E over M. A horizontal bundle is then a particular choice of a subbundle of TE which is complementary to VE, in other words provides a complementary subspace in each fiber.
In full generality, the horizontal bundle concept is one way to formulate the notion of an Ehresmann connection
on a fiber bundle
. However, the concept is usually applied in more specific contexts.
More precisely, if e ∈ E with
then the vertical space VeE at e is the tangent space Te(Ex) to the fiber Ex through e. A horizontal bundle then determines an horizontal space HeE such that TeE is the direct sum of VeE and HeE.
If E is a principal G-bundle
then the horizontal bundle is usually required to be G-invariant: see Connection (principal bundle)
for further details. In particular, this is the case when E is the frame bundle
, i.e., the set of all frames for the tangent spaces of the manifold, and G = GLn.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, in the field of differential topology
Differential topology
In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.- Description :...
, given
- π:E→M,
a smooth fiber bundle
Fiber bundle
In mathematics, and particularly topology, a fiber bundle is intuitively a space which locally "looks" like a certain product space, but globally may have a different topological structure...
over a smooth manifold M, then the vertical bundle
Vertical bundle
The vertical bundle of a smooth fiber bundle is the subbundle of the tangent bundle that consists of all vectors which are tangent to the fibers...
VE of E is the subbundle of the tangent bundle
Tangent bundle
In differential geometry, the tangent bundle of a differentiable manifold M is the disjoint unionThe disjoint union assures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector...
TE consisting of the vectors which are tangent to the fibers of E over M. A horizontal bundle is then a particular choice of a subbundle of TE which is complementary to VE, in other words provides a complementary subspace in each fiber.
In full generality, the horizontal bundle concept is one way to formulate the notion of an Ehresmann connection
Ehresmann connection
In differential geometry, an Ehresmann connection is a version of the notion of a connection, which makes sense on any smooth fibre bundle...
on a fiber bundle
Fiber bundle
In mathematics, and particularly topology, a fiber bundle is intuitively a space which locally "looks" like a certain product space, but globally may have a different topological structure...
. However, the concept is usually applied in more specific contexts.
More precisely, if e ∈ E with
- π(e)=x ∈ M,
then the vertical space VeE at e is the tangent space Te(Ex) to the fiber Ex through e. A horizontal bundle then determines an horizontal space HeE such that TeE is the direct sum of VeE and HeE.
If E is a principal G-bundle
Principal bundle
In mathematics, a principal bundle is a mathematical object which formalizes some of the essential features of the Cartesian product X × G of a space X with a group G...
then the horizontal bundle is usually required to be G-invariant: see Connection (principal bundle)
Connection (principal bundle)
In mathematics, a connection is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points...
for further details. In particular, this is the case when E is the frame bundle
Frame bundle
In mathematics, a frame bundle is a principal fiber bundle F associated to any vector bundle E. The fiber of F over a point x is the set of all ordered bases, or frames, for Ex...
, i.e., the set of all frames for the tangent spaces of the manifold, and G = GLn.