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