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Let $X$ be a smooth vector field.
Let $\omega$ be a smooth covector field.
Let $X^\flat$ and $\omega^\sharp$ denote the index lowering and index raising of $X$ and $\omega$ respectively.
Then the mappings $\flat$ and $\sharp$ are called the musical isomorphisms.
- 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Basic Constructions on Riemannian Manifolds