Definition:Connection on Manifold

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

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

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:


 * $\ds \nabla_{f_1 X_1 + f_2 X_2} Y = f_1 \nabla_{X_1} Y + f_2 \nabla_{X_2} Y$


 * $\ds \map {\nabla_X} {a_1 Y_1 + a_2 Y_2} = a_1 \nabla_X Y_1 + a_2 \nabla_X Y_2$


 * $\ds \map {\nabla_X} {f Y} = f \nabla_X Y + \paren {X f} Y$

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