What was the motivation for the development of modern, intrinsic, differential geometry?

Question

I know that tensor calculus was developed around the same time as general relativity. Tensor calculus was the prime way to deal with geometric objects, based on expliciting all coordinates and doing computations with them (the so called "debauch of indices").

Modern differential geometry is much more concerned with intrinsic properties, set on stages called "manifolds". How did this happen? What motivated them to develop these entities? Was the gain immediately noticeable compared to the jump in abstraction involved?

Answer 1

Tensor calculus was developed about 20 years before the general relativity by Ricci and Levi-Civita, starting around 1890, under the name of absolute differential calculus. It was motivated by Riemann's work on manifolds with a metric, and summarized in their comprehensive 1900 book. Einstein learned about it from a geometer friend Grossman around 1912, and saw it as a good vehicle to express his new ideas for geometrizing mechanics. It took him two years of correspondence with Levi-Civita to master the techniques. He even contributed some simplifications like the use of upper and lower indices, and the summation convention. However, coordinate tensor calculus was in principle at odds with Einstein's ideas about relativity and general covariance since it relied on local coordinates.

The invariant turn is associated with the work of Elie Cartan in 1920-s on integrability of so-called Pfaffian systems, where he developed differential forms as an alternative coordinate free notation. In 1950 Kozsul used it to eliminate non-tensorial objects like Christoffell symbols from the theory of covariant differentiation, and swift adoption of differential forms by many mathematicians followed in 1950-s. It became a standard due to the Bourbaki project, the expression "debauchery of indices" is theirs, and it tells us how they felt about the coordinate calculus.

Part of it is that coordinate free notation is more "natural" because there is no global choice of coordinates on manifolds. Part of it is algebraic elegance that allows for cleaner and more conceptual proofs in geometry. Another part is that it fit very well with the framework of algebraic topology, intensively pursued in 1950-s, particularly with deRham cohomology and the theory of characteristic classes. And part of it, especially later, is the strong influence of Bourbaki on standardization of mathematics in general.

Physicists, who are more often interested in computations than in proofs, were never fully sold on the invariant notation, and continue to use indices alongside it. However, after the publication of Yang-Mills paper in 1954, and the realization that gauge fields are intrinsically connection forms already studied by mathematicians, many physicists invested into learning both languages to understand their counterparts. Also, the ideas of gauge invariance and general covariance are more manifestly expressed by invariant concepts and techniques.

Answer 2

It begins with Riemann's lecture in Göttingen in 1854, Über die Hypothesen welche der Geometrie zu Grunde liegen". Here is an English translation:

http://www.emis.de/classics/Riemann/WKCGeom.pdf

As Paul Siegel said, Gauss theory of surfaces and Lobachevski geometry gave some motivation. As another motivation, one can mention Riemann's work on in mathematical physics (heat conduction in solids).

I mean his work titled "A mathematical work that seeks to answer the question posed by the most distinguished academy of Paris", number XXII in his Collected papers. This work is a predecessor of the tensor analysis.

Briefly, the motivation is the following: when you consider a surface, you can always imagine it embedded, or immersed, at least locally. But with 3-D and higher dimensional spaces this does not help much, so one naturally starts to consider intrinsic geometry. One early example of this is Lobachevski geometry in dimension 3.

In turn, motivation for the Gauss theory of surfaces was geodesy.

Answer 3

I don't have references on hand, but I believe intrinsic geometry began with Gauss' Theorema Egregium, which asserts that the Gaussian curvature of a surface at a point depends only on the intrinsic metric in a neighborhood of that point and not on the embedding of the surface (in other words, it is invariant under local isometry). This is a bit surprising because the principal curvatures (which are measured along curves in the surface) depend very much on the embedding. Gauss' observation suggests that it is possible - and desirable - to have a language for describing the geometry of surfaces without reference to a chosen embedding. I think this helped motivate the development of the metric and curvature tensors.

Hyperbolic geometry also plays an important role in this story. The parallel postulate was very much on Gauss' mind when he was thinking about curvature and the geometry of surfaces, and there is evidence that he wrote down the axioms of hyperbolic geometry several years before Lobachevsky but decided not to publish it. Most likely he kept it secret because he could not construct a model for the hyperbolic plane and thus he was not convinced that he had shown that non-Euclidean geometry "exists" according to the standards of the time. The problem is that one cannot embed the hyperbolic plane in $\mathbb{R}^3$, though one can in a sense do so locally; thus one needs a geometric language in which a global object is pieced together from well-understood local objects. The theory of Riemann surfaces was a first step in this direction, but perhaps it was not until the language of Riemannian manifolds was invented that Euclidean geometry and hyperbolic geometry were truly placed on the same footing.