In Riemannian geometry, the "lowering indices" operator is denoted by $\flat:TM \to T^*M$ and the "raising indices" operator by $\sharp:T^*M \to TM$. These isomorphisms are sometimes referred to as musical isomorphisms, as stated on Wikipedia and in several other sources. Surely, the motivation for such terminology is clear. I would nevertheless like to know who decided to adpot these (rather amusing) notations, so here is a question:

What was the first paper / textbook that made use of the notations $\flat$ and $\sharp$?

and a possible follow-up question:

If such notations were not adopted widely after the first appearance, who popularized them?

  • 3
    $\begingroup$ Wasn't it Marcel Berger who introduced them? I might be wrong... $\endgroup$ – Olivier Bégassat Jun 29 '11 at 1:49

Marcel Berger, on p. 696 of his Springer-book: A panoramic view of Riemannian Geometry, writes "Next we define the canonical musical duality", referring to a footnote: "These dualities are called musical because they are often written with symbols like ♭: V $\rightarrow$ V* and ♯: V* $\rightarrow$ V."

Even if this is not the first metioning of "musical", we can assume that Berger invented that expression, because in a panoramic viev of over 800 pages, a reference to another originator could have been expected.

Further Rob Kusner's hint here # or "sharp" is what Marcel Berger calls a "musical isomorphism" and Olivier Bégassat's comment here seem to provide sufficient evidence to determine the inventor.

To answer this question completely, one would have to read through Berger's complete works. But if we wait for a while, this task will probably be reduced to touching a button.


The musical isomorphisms already appear in 1971 on page 21 of BERGER M. et al, Le spectre d'une variete riemannienne, Lecture Notes in Math. 194, 1971, Springer, see http://ci.nii.ac.jp/naid/10003477917/

However, I am not convinced that this is the first place where they appear, or that Berger was the inventor. One would have to trace German textbooks in Riemannian geometry from the 1960s or perhaps earlier. Interesting question!

  • 2
    $\begingroup$ Maybe someone could ask Berger directly? $\endgroup$ – Deane Yang Apr 11 '13 at 12:56
  • 9
    $\begingroup$ I did, but he does not remember. $\endgroup$ – Mikhail Katz Apr 13 '13 at 20:12

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.