Sharp map

Short description: Mathematical technique

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

$X^{\mathrm{♭}}:=i_{X}g=g(X,\cdot )$

yields a linear map sometimes called the flat map

$\mathrm{♭}:\mathrm{\Gamma }(TM)\to \mathrm{\Gamma }(T^{*}M)$

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

$\mathrm{♯}:={\mathrm{♭}}^{-1}:\mathrm{\Gamma }(T^{*}M)\to \mathrm{\Gamma }(TM)$

is called the sharp map.