# Role of Group actions in Differential Geometry

This is a rather soft question, my hope is to bring some order into the stuff I would like to learn about differential geometry -- here it is:

I was told over and over again that Geometry has to do with angles, length of curves, curvature etc.

However, most graduate textbooks on Differential Geometry start with smooth manifolds and keep studying them without every mentioning the presence of a Riemannian metric. And then there are group actions acting on smooth manifolds -- here I see the use of the word "Geometry" as well when results are obtained regarding the basic structure of the manifold, such as the way orbits are arranged, whether there are special orbits and how orbits smoothly collapse, whether these are certain bundles and so on.

I guess my question is the following: Is the study of group actions on smooth manifolds "more fundamental" than Riemannian geometry, or can one reduce any group action on a manifold to the presence of a (unique?) Riemannian metric on $M$ and study that metric instead? If not, are there "geometric" aspects of a smooth manifold that one cannot frame in terms of a group action?

(Maybe I should add that this question is motivated in the context of real (as opposed to complex) smooth manifolds)

Thanks for sharing your insight!

• In the sense that you're asking, it seems fair to say group actions are "more fundamental" than Riemannian geometry. There's a viewpoint from which a "geometric" concept (e.g., straightness, angle, length) is precisely an invariant of some group action. The Wikipedia page on Klein's Erlangen Program might be a good starting point. Kobayashi's Transformation Groups in Differential Geometry may also be of interest. – Andrew D. Hwang Mar 12 '15 at 14:05

## 1 Answer

It is wrong to say that the study of group actions is "more fundamental" than Riemannian geometry, though I might secretly harbor that contention.

Riemannian geometry can be interpreted as the study of $O_n$-structures. A choice of Riemannian metric is equivalent to a choice of $O_n$-structure.

There are, however, transformation groups that don't correspond to a metric. For example, the diffeomorphism group of a smooth manifold has nothing to do with a metric.

A plethora of geometric structures can be interpreted as reductions of structure groups. Transformation groups are pretty awesome that way.