| Musical Isomorphism |
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| CATEGORIES ABOUT MUSICAL ISOMORPHISM | |
| riemannian geometry | |
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INTRODUCTION A metric ''g'' on a Riemannian manifold ''M'' is a Tensor Field which is Symmetric , Nondegenerate and Positive-definite . If we fix one parameter as a vector , we have an isomorphism of vector spaces:
given by : i.e. : Globally the map
is a Diffeomorphism . MOTIVATION OF THE NAME The isomorphism and its inverse are called ''musical isomorphisms'' because they move up and down the indexes of the vectors. For instance, a vector of ''TM'' is written as and a covector as , so the index ''i'' is moved up and down in just as the symbols Sharp () and Flat () move up and down the Pitch of a Tone . GRADIENT The musical isomorphisms can be used to define the Gradient of a Smooth Function over a Riemannian manifold ''M'' as follows: : |
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