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

${\displaystyle X^{\flat }:=i_{X}g=g(X,.)}$

yields a linear map sometimes called the flat map

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

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

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

is called the sharp map.