Sharp map

In differential geometry, the sharp map is the mapping that converts coordinate 1-forms into corresponding coordinate basis vectors.

Definition

Let $M$ be a manifold and $\,\Gamma (TM)$ denote the space of all sections of its tangent bundle. Fix a nondegenerate (0,2)-tensor field $g \in \Gamma(T^*M^{\otimes 2})$ , i.e., a metric tensor or a symplectic form. The definition

$X^\flat := i_X g = g(X,.)$

yields a linear map sometimes called the flat map

$\flat : \Gamma(TM) \to \Gamma(T^*M)$

which is an isomorphism, since $g$ is non-degenerate. Its inverse

$\sharp := \flat^{-1} : \Gamma(T^*M) \to \Gamma(TM)$

is called the sharp map.