Connection on a vector bundle
This lives as an element of: the space of -bilinear maps for a vector bundle over a manifold
This article defines a basic construct that makes sense on any differential manifold
View a complete list of basic constructs on differential manifolds
Definition part (pointwise form)
A connection is a smooth choice of the following: at each point , there is a map , satisfying some conditions. The map is written as where and .
- It is -linear in (i.e., in the coordinate).
- It is -linear in (viz., the space of sections on ).
- It satisfies the following relation called the Leibniz rule:
Definition part (global form)
A connection is a map , satisfying the following:
- It is -linear in (in other words, it is tensorial, or pointwise, in the -coordinate)
- it is -linear in
- It satisfies the following relation called the Leibniz rule:
where is a scalar function on the manifold and denotes scalar multiplication of by .
A connection is equivalent to the following:
- A choice of splitting of the first-order symbol sequence of a vector bundle: Further information: connection is splitting of first-order symbol sequence
- A module structure of the vector bundle, over the connection algebra: Further information: connection is module structure over connection algebra
- A connection on the corresponding bundle over the principal bundle over the general linear group. Further information: connection on vector bundle equals connection on principal GL-bundle
When is the trivial one-dimensional bundle, then sections of are the same as infinitely differentiable functions on . For this bundle, there is a unique connection: the usual action of a vector field on a function.
When is itself the tangent bundle, we call the connection a linear connection.
Covariant derivative of a section
Further information: covariant derivative of a section
The term covariant derivative can thus be used only if we already have a connection in mind.
Absolute derivative of a section
Further information: absolute derivative of a section
Given a connection , the absolute derivative of a section , denoted , is defined as the operator that sends a vector field to . In the particular case where is the trivial one-dimensional bundle, this reduces to the de Rham derivative of a function, yielding a 1-form.
Connection, transport along a curve
Given a connection on the manifold, we can obtain a connection along any curve on the manifold, using the pullback connection. A connection along the curve gives a transport: a rule for transporting a basis for the fiber at one point, to a basis for the fiber at the other point. Thus, a connection is often thought of as a global transport rule.
Consider a vector field on . We know that we can define a notion of directional derivatives for functions along this vector field: this differentiates the function at each point, along the vector at that point. The derivative of along the direction of is a new function, denoted as .
Note that at any point , the value of depends on the local behavior of but only on the pointwise behavior of , that is, it only depends on the tangent vector and not on the behavior of in the neighborhood.
The idea behind a connection is to extend this differentiation rule, not just to functions, but also to other kinds of objects. In particular, we want to be able to have a differentiation rule for sections of the tangent and cotangent bundles, along vector fields. In this definition, what we would like is:
- The derivative with respect to a vector field at a point should just depend on the value of the vector field at the point -- it should not depend on the behavior in the neighborhood. We say it is a tensorial map with respect to .
- A Leibniz rule is satisfied with respect to scalar multiplication by functions, which connects differentiation for this connection with the differentiation of scalar functions along vector fields
Note that the usual differentiation along vector fields is thus the canonical connection on the trivial one-dimensional bundle, and we would like that any other connections we define should be compatible with this via the Leibniz rule.
Further information: Connections exist
Connection on a direct sum
Further information: Direct sum of connections
Connection on a tensor product
Further information: Tensor product of connections
In other words, the formula is chosen so that a Leibniz-like rule is satisfied for tensor products.
Connection on the dual
Further information: Dual connection
Given a connection on a vector bundle over a differential manifold , we can obtain a connection on the dual bundle as follows:
Particular kinds of connections
Further information: metric connection
The notion of a metric connection makes sense when we have a metric bundle: a vector bundle with an inner product on every fiber that varies compatibly. A metric connection is a connection with the property that it satisfes a Leibnixz-like rule with respect to the inner product of sections:
The set of all connections
As an affine space
Further information: Affine space of all connections Given a manifold and a vector bundle over , consider the set of all connections for . Clearly, the connections live inside the space of -bilinear maps . Hence, we can talk of linear combinations of connections. In general, a linear combination of connections need not be a connection. The problem arises from the Leibniz rule, which has a term that does not scale with the connection.
It is true that the set of differences of connections (if nonempty) forms a vector subspace of the vector space of all bilinear maps. Since there is a fundamental theorem that connections exist, we conclude that the set of connections is in fact an affine space, viz a translate of a subspace, and thus any affine linear combination of connections is again a connection.
As the collection of module structures
Further information: Connection is module structure over connection algebra Given a vector bundle , a connection on makes act on . Thus, we could view as a module over the free algebra generated by . This action actually satisfies some extra conditions, and these conditions help us descend to an action of the connection algebra on .
Thus, a connection on a vector bundle is equivalent to equipping with a module structure over the connection algebra.
Further information: Connections localize
Given a connection on the whole differential manifold , we can get a connection on any open subset of . Note that this is not a completely trivial statement, because not every vector field on an open subset extends to a vector field on the whole manifold. However, we can express any vector field on an open subset, as the product of a function and a vector field that can be extended to the whole manifold, and we can then use the Leibniz rule.
It is also true that connections piece together. In other words, to know at a point , it suffices to know the germ of at .
Describing connections using coordinate charts
Further information: Christoffel symbols of a connection, matrix of connection forms A connection is a bilinear map, and because connections localize and piece together, it suffices to describe what happens to the connection inside coordinate charts. However, we need to remember that while depends only pointwise on (so it depends only on the value of at a point), it depends locally on (so it depends on the germ of at the point). So, to describe a connection at a point , it is not enough to take a basis for and a basis for and describe what happens on that basis. Rather, we take a basis for , and pick a coordinate chart around , and take constant vector fields corresponding to a choice of basis for that coordinate chart.