In the mathematical field of

differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...

, the term linear connection can refer to either of the following overlapping concepts: * a

connection on a vector bundle In mathematics, and especially differential geometry and gauge theory, a connection on a fiber bundle 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. Th ...

, often viewed as a differential operator (a ''Koszul connection'' or ''

covariant derivative In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a different ...

''); * a

principal connection In mathematics, and especially differential geometry and gauge theory, 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. A principal ''G''-conne ...

on the

frame bundle In mathematics, a frame bundle is a principal fiber bundle F(''E'') associated to any vector bundle ''E''. The fiber of F(''E'') over a point ''x'' is the set of all ordered bases, or ''frames'', for ''E'x''. The general linear group acts natur ...

of a manifold or the induced connection on any

associated bundle In mathematics, the theory of fiber bundles with a structure group G (a topological group) allows an operation of creating an associated bundle, in which the typical fiber of a bundle changes from F_1 to F_2, which are both topological spaces wit ...

— such a connection is equivalently given by a

Cartan connection In the mathematical field of differential geometry, a Cartan connection is a flexible generalization of the notion of an affine connection. It may also be regarded as a specialization of the general concept of a principal connection, in which the ...

for the

affine group In mathematics, the affine group or general affine group of any affine space over a field is the group of all invertible affine transformations from the space into itself. It is a Lie group if is the real or complex field or quaternions. Relat ...

affine space In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relate ...

, and is often called an

affine connection In differential geometry, an affine connection is a geometric object on a smooth manifold which ''connects'' nearby tangent spaces, so it permits tangent vector fields to be differentiated as if they were functions on the manifold with values i ...

. The two meanings overlap, for example, in the notion of a linear connection on the

tangent bundle In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...

of a manifold. In older literature, the term ''linear connection'' is occasionally used for an

Ehresmann connection In differential geometry, an Ehresmann connection (after the French mathematician Charles Ehresmann who first formalized this concept) is a version of the notion of a connection, which makes sense on any smooth fiber bundle. In particular, it d ...

on an arbitrary fiber bundle, to emphasise that these connections are "linear in the horizontal direction" (i.e., the

horizontal bundle In mathematics, the vertical bundle and the horizontal bundle are vector bundles associated to a smooth fiber bundle. More precisely, given a smooth fiber bundle \pi\colon E\to B, the vertical bundle VE and horizontal bundle HE are subbundles of ...

is a ''vector'' subbundle of the tangent bundle of the fiber bundle), even if they are not "linear in the vertical (fiber) direction". However, connections which are not linear in this sense have received little attention outside the study of spray structures and

Finsler geometry In mathematics, particularly differential geometry, a Finsler manifold is a differentiable manifold where a (possibly asymmetric) Minkowski functional is provided on each tangent space , that enables one to define the length of any smooth c ...

References

{{reflist Connection (mathematics)