Sharp map

In differential geometry, the sharp map is the mapping that converts 1-forms into corresponding vectors, given a non-degenerate (0,2)-tensor.

Definition

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

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

yields a linear map sometimes called the flat map

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

which is an isomorphism, since ${\displaystyle g}$ is non-degenerate. Its inverse

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

is called the sharp map.