# Definition:Connection on Manifold

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## Definition

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

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

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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 $M$**.

Although this article appears correct, it's inelegant. There has to be a better way of doing it.In particular: Can we refer to Koszul connection even more to simplify this page further?You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by redesigning it.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Improve}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

## Also see

- Results about
**connections**can be found**here**.

## Sources

- 2018: John M. Lee:
*Introduction to Riemannian Manifolds*(2nd ed.) ... (previous) ... (next): $\S 4$: Connections. Connections in the Tangent Bundle