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It is also known as Raising And Lowering Indices .


INTRODUCTION

  • M\, of symmetric bilinear forms on the tangent bundle. At any point ''x''∈''M'', g_x\in S^2T^---_xM defines an isomorphism of vector spaces

  • }_x M

    • (from the tangent space to the cotangent space) given by

:\widehat{g}_x(X_x) = g(X_x,-)
for any tangent vector ''X''''x'' in ''T''''x''M, i.e.,
: \widehat{g}_x(X_x)(Y_x) = g_x(X_x,Y_x).

The collection of these linear isomorphisms define a bundle isomorphism
  • }M

    • which is therefore, in particular, a Diffeomorphism . This is called the musical isomorphism '''''flat''''', and its inverse is called '''''sharp''''': sharp raises indices, flat lowers them.



MOTIVATION OF THE NAME

The isomorphism \widehat{g} and its inverse \widehat{g}^{-1} are called ''musical isomorphisms'' because they move up and down the indexes of the vectors. For instance, a vector of ''TM'' is written as \alpha^i rac{\partial}{\partial x^i} and a covector as \alpha_i dx^i, so the index ''i'' is moved up and down in \alpha just as the symbols Sharp (\sharp) and Flat ( lat) move up and down the Pitch of a Semitone .


GRADIENT

The musical isomorphisms can be used to define the Gradient of a Smooth Function over a Riemannian manifold ''M'' as follows:

:\mathrm{grad}\;f=\widehat{g}^{-1} \circ df = (df)^{\sharp}