Musical isomorphism

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In mathematics, the musical isomorphism (or canonical isomorphism) is an isomorphism between the tangent bundle TM and the cotangent bundle T
 
M
of a pseudo-Riemannian manifold induced by its metric tensor. There are similar isomorphisms on symplectic manifolds. The term musical refers to the use of the symbols (flat) and (sharp).[1][2]

It is also known as raising and lowering indices.

Discussion[edit]

Let (M, g) be a pseudo-Riemannian manifold. Suppose {ei} is a basis for the tangent bundle TM with dual basis {ei}. Then, locally, we may express the pseudo-Riemannian metric (which is a 2-covariant tensor field that is symmetric and nondegenerate) as g = gijeiej (where we employ the Einstein summation convention). Given a vector field X = Xiei we define its flat by

This is referred to as "lowering an index". Using the traditional diamond bracket notation for inner product defined by g, we obtain the somewhat more transparent relation

for all vectors X and Y.

Alternatively, given a covector field ω = ωiei we define its sharp by

where gij are the elements of the inverse matrix to gij. Taking the sharp of a covector field is referred to as "raising an index". In inner product notation, this reads

for all covectors ω and vectors Y.

Through this construction we have two mutually inverse isomorphisms

These are isomorphisms of vector bundles and hence we have, for each p in M, mutually inverse vector space isomorphisms between TpM and T
p
M
.

Extension to tensor products[edit]

The musical isomorphisms may also be extended to the bundles

Which index is to be raised or lowered must be indicated. For instance, consider the (0, 2) tensor field X = Xijeiej. Raising the second index, we get the (1, 1) tensor field

Extension to k-vectors and k-forms[edit]

In the context of exterior algebra, an extension of the musical operators may be defined on V and its dual
 
V
, which with minor abuse of notation may be denoted the same, and are again mutual inverses:[3]

defined by

In this extension, in which maps p-vectors to p-covectors and maps p-covectors to p-vectors, all the indices of a totally antisymmetric tensor are simultaneously raised or lowered, and so no index need be indicated:

Trace of a tensor through a metric tensor[edit]

Given a type (0, 2) tensor field X = Xijeiej, we define the trace of X through the metric tensor g by

Observe that the definition of trace is independent of the choice of index to raise, since the metric tensor is symmetric.

See also[edit]

Notes[edit]

  1. ^ Lee 2003, Chapter 11.
  2. ^ Lee 1997, Chapter 3.
  3. ^ Vaz, Jr., Jayme; Roldão, Jr., da Rocha (2016). An Introduction to Clifford Algebras and Spinors. Oxford University Press. 

References[edit]

  • Lee, J. M. (2003). Introduction to Smooth manifolds. Springer Graduate Texts in Mathematics. 218. ISBN 0-387-95448-1. 
  • Lee, J. M. (1997). Riemannian Manifolds – An Introduction to Curvature. Springer Graduate Texts in Mathematics. 176. New York · Berlin · Heidelberg: Springer Verlag. ISBN 978-0-387-98322-6.