Definition:Koszul Connection

Definition

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 $E$.

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Also see

• Results about connections can be found here.

Source of Name

This entry was named for Jean-Louis Koszul.

Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 4$: Connections. Connections