From a perspective of math history, what discoveries resulted directly from Riemann’s discovery and codification of non-flat metrics on Riemannian manifolds? After Albert Einstein modeled the Universe on a pseudo-Riemannian manifold, Riemann’s work was rediscovered. But Einstein was a physicist. Math concepts like the Cartan connection and bundle and fiber model for the tangent space seem related to curved metrics, but did not emerge from it directly.
One motivation for this question is the fact that a curved metric does not change the result for total arc length when integrated compared to the Euclidean one from which it is derived (for simple real curves). That made me think there must be some other important significance to the idea that the distance between nearby points can be something other than straight. A non-flat metric seems to be such a cornerstone of Riemannian geometry.
As an example of arc length invariance, see https://math.stackexchange.com/questions/4441646/do-the-curvature-properties-of-exotic-spheres-result-in-necessary-new-techniques . The circumference of a great circle on a sphere is the same as that of a circle produced by a planar intersection at the equator (this also applies to curves on higher dimensional manifolds).
The question arises: what calculations did change directly as a result of the fact that curved spaces have non-flat metrics?
The following things did result from Riemann’s overall body of work on Riemannian manifolds:
- Established that many geometries exist
- That Riemann curvature and metric are intrinsic to a surface, the same way Gaussian curvature is.
- It led to new branches of mathematics such as algebraic geometry, the Cartan and other connections, and the bundle and fiber concepts of the tangent space.
None of these proceeds directly from the fact that a metric itself can be conceived of as curved, geometrically. It seems to me that there must have been something additional that resulted. For example, did Riemann’s work enable arc lengths on manifolds to be calculated simply with a single integral of his metric? Using a Euclidean metric on portions of a curve on a curved space would be tedious, as each planar intersection segment (especially for a higher dimensional manifold) would have to be added to others to arrive at total arc length. Geometers other than Riemann in 1860 didn’t spend time on curves on higher dimensional manifolds, rendering my suggestion sort of moot. But if they had...
Or did the aforementioned discovery have a direct result on the calculation of complex curves - curves in complex spaces that use the imaginary number i? I mean would arc lengths not be the same for the curve on the manifold compared to that on a planar (or several) intersection(s)? This is an area I know nothing about, and requires expert mathematical advice to answer. I would obviously have to define which axes are imaginary and which are real first.
Is it better to view the concept of a curved metric simply as part of the greater picture, and not separated – put in context along with curved spaces, the curvature tensor, Christoffel symbols, and higher dimensional shapes? Then answer the question of how these changed the course of mathematics development.