In this article in Wikipedia about Katsumi Nomizu https://en.wikipedia.org/wiki/Katsumi_Nomizu it is written that

"Over the course of his career, Katsumi Nomizu was influential in determining the course of differential geometry by stressing what he called the structural approach."

In what does this structural approach constist? Are there some papers written by Nomizu where this concept is clarified?


1 Answer 1


The intended sense of "structure" comes from Bourbaki, who reformulated all of mathematics as a theory of structures founded on set theory in their multi-volume Elements of Mathematics (with the obvious allusion to Euclid), see Nicolas Bourbaki: Theory of Structures. A structure is a set with operations and/or relations defined on it and/or additional objects attached to it. Topology, metric, group operation, partial order, etc., are examples of structures.

In differential geometry the basic structure is the differential one, and one can put extras on top of it, affine connection, Riemannian metric, symplectic form, complex or Kähler structure, etc. Despite having prominent geometers among its members and decrying the "debauchery of indices" in classical expositions, see What was the motivation for the development of modern, intrinsic, differential geometry?, Bourbaki never quite got to writing its own volume on differential geometry.

Kobayashi-Nomizu's Foundations of Differential Geometry became the stand-in for that. It promoted systematic use of invariant notation and intrinsic descriptions of geometric objects through the systematic use of the language of Lie groups, homogeneous spaces, fiber bundles, connections on bundles, and other differential geometric structures. Nomizu particularly contributed to the revival of interest in affine structures in 1980-s with his Munster lecture What is Affine Differential Geometry? and subsequent work, see Geometry And Topology Of Submanifolds VII, p.35.

  • $\begingroup$ I have this feeling that the field of Differential Geometry in these years is being absorbed into Geometric Analysis. And structure is suffering because of this passage from people that had a formation based more on Algebraic Geometry or Complex Analysis to people that came from Analysis. I see a lot of efforts spent in find solutions of PDE systems in order to find metrics that satisfies some constraint, while the first questions should be about the classification of the objects (e.g. finding invariants) in the studied category. What do you think ? $\endgroup$ Commented Jan 30, 2018 at 21:56
  • $\begingroup$ @WarlockofFiretopMountain Perhaps, but there is a reason for that. Donaldson, then Floer, Taubes and recently Perelman showed that geometric analysis can solve classificatory problems that classical geometric methods could not handle. Even the geometer's geometer, Gromov, introduced pseudoholomorphic curves to define new invariants. And string theory demands either algebraic geometry or (symplectic) geometric analysis. People gravitate to where hot action is, compactified moduli invariants and Ricci flow are the new surgery, and the technical overflow is perhaps an excess of a growth spurt. $\endgroup$
    – Conifold
    Commented Jan 31, 2018 at 3:29

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