Definition:Connection

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Definition

Let $M$ be a differentiable manifold.

A connection on $M$ is a way of defining the parallelism of vectors.

It allows consistent differentiation everywhere on $M$, as it "connects" the various local coordinates of $M$.


Connection on Manifold

Let $M$ be a smooth manifold with or without boundary.

Let $\map {\mathfrak X} M$ be the space of smooth vector fields on $M$.

Let $\map {C^\infty} M$ be the space of smooth real functions on $M$.

Let $\nabla : \map {\mathfrak X} M \times \map {\mathfrak X} M \to \map {\mathfrak X} M$ be the map be written $\tuple {X, Y} \mapsto \nabla_X Y$ where $X, Y \in \map {\mathfrak X} M$, and $\times$ denotes the cartesian product.


Suppose $\forall f, f_1, f_2 \in \map {C^\infty} M$ and $\forall a_1, a_2 \in \R$ we have that $\nabla$ satisfies the following:

$\nabla_{f_1 X_1 + f_2 X_2} Y = f_1 \nabla_{X_1} Y + f_2 \nabla_{X_2} Y$
$\map {\nabla_X} {a_1 Y_1 + a_2 Y_2} = a_1 \nabla_X Y_1 + a_2 \nabla_X Y_2$
$\map {\nabla_X} {f Y} = f \nabla_X Y + \paren {X f} Y$


Then $\nabla$ is known as the connection on $M$.


Koszul Connection

Let $M$ be a smooth manifold with or without boundary.

Let $E$ be a smooth manifold.

Let $\pi : E \to M$ be a smooth vector bundle.

Let $\map \Gamma E$ be the space of smooth sections of $E$.


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Let $\map {\mathfrak{X}} M$ be the space of smooth vector fields on $M$.


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Let $\map {C^\infty} M$ be the space of smooth real functions on $M$.

Let $\nabla : \map {\mathfrak{X}} M \times \map \Gamma E \to \map \Gamma E$ be the map be written $\tuple {X, Y} \mapsto \nabla_X Y$ where $X \in \map {\mathfrak{X}} M$, $Y \in \map \Gamma E$, and $\times$ denotes the cartesian product.

Suppose $\nabla$ satisfies the following:

\(\ds \nabla_{f_1 X_1 + f_2 X_2} Y\) \(=\) \(\ds f_1 \nabla_{X_1} Y + f_2 \nabla_{X_2} Y\)
\(\ds \map {\nabla_X} {a_1 Y_1 + a_2 Y_2}\) \(=\) \(\ds a_1 \nabla_X Y_1 + a_2 \nabla_X Y_2\)
\(\ds \map {\nabla_X} {f Y}\) \(=\) \(\ds f \nabla_X Y + \paren {X f} Y\)

where:

$f, f_1, f_2 \in \map {C^\infty} M$
$a_1, a_2 \in \R$


Then $\nabla$ is known as the connection in $E$.


Levi-Civita Connection

Let $\struct {M, g}$ be a Riemannian or pseudo-Riemannian manifold with or without boundary.

Let $TM$ be the tangent bundle of $M$.

Let $\nabla$ be a connection on $TM$.


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Suppose $\nabla$ is compatible with $g$ and symmetric.


Then $\nabla$ is called Levi-Civita connection.


Also see

  • Results about connections can be found here.


Historical Note

The idea of a connection was investigated by Hermann Klaus Hugo Weyl in his efforts to unify the theory of relativity and that of electromagnetism.

Subsequently, connections were used in the study of vector bundles, and as a result have shown themselves to be basic to the development of gauge theory.


Sources

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