Definition:Connection Difference Tensor

Definition

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

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

Let $\nabla^0$ and $\nabla^1$ be any two connections in $TM$.

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

Let $D : \map {\mathfrak{X}} M \times \map {\mathfrak{X}} M \to \map {\mathfrak{X}} M$ the mapping such that:

$\ds \forall X, Y \in \map {\mathfrak{X}} M : \map D {X, Y} := \nabla^1_X Y - \nabla^0_X Y$

where $\times$ denotes the cartesian product and $\nabla_X Y$ is the covariant derivative of $Y$ in the direction $X$.

Then $D$ is called the connection difference tensor.

Sources

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