Definition:Curl Operator/Riemannian Manifold

Definition

Let $\struct {M, g}$ be a Riemannian manifold.

Let $TM$ and $T^*M$ be the tangent bundle and the cotangent bundle of $M$ respectively.

Let $\wedge^2 T^*M$ be the subbundle of alternating tensors.

Let $\rd V_g$ be the Riemannian volume form.

Let $\beta : TM \to \wedge^2 T^*M$ be a mapping such that:

$\map \beta X = X \lrcorner \rd V_g$

where $\lrcorner$ denotes the interior multiplication.

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

Let $X \in \map {\mathfrak{X}} M$ be a smooth vector field.

Let $\flat$ denote the index lowering.

Then the curl operator is defined by:

$\curl X := \beta^{-1} \map d {X^\flat}$

Sources

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