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 be a manifold and denote the space of all sections of its tangent bundle. Fix a nondegenerate (0,2)-tensor field , for example a metric tensor or a symplectic form. The definition

yields a linear map sometimes called the flat map

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

is called the sharp map.

See also


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