Table of Contents

## When did the theory of differential geometry develop?

Since the late 1940s and early 1950s, differential geometry and the theory of manifolds has developed with breathtaking speed. It has become part of the ba- sic education of any mathematician or theoretical physicist, and with applications in other areas of science such as engineering or economics.

## Which is the fundamental object of differential geometry?

In the words of S.S. Chern, ”the fundamental objects of study in differential geome- try are manifolds.”1Roughly, an n-dimensional manifold is a mathematical object that “locally” looks like Rn. The theory of manifolds has a long and complicated history.

**Why are level sets important in differential geometry?**

Quite possibly, one reason is that for quite a while, the concept as such was mainly regarded as just a change of perspective (away from level sets in Eu- clidean spaces, towards the ‘intrinsic’ notion of manifolds).

**Which is an example of a sub branch of differential geometry?**

There are many sub- branches, for example complex geometry, Riemannian geometry, or symplectic ge- ometry, which further subdivide into sub-sub-branches. 3See e.g. the article by Scholz http://www.maths.ed.ac.uk/ aar/papers/scholz.pdf for the long list of names involved.

### Which is an example of a manifold in differential geometry?

For centuries, manifolds have been studied as subsets of Euclidean space, given for example as level sets of equations.

### Why was gauge theory important to differential geometry?

Some years later, gauge theory once again emphasized coordinate-free formulations, and provided physics motivations for more elaborate constructions such as ﬁber bundles and connections. Since the late 1940s and early 1950s, differential geometry and the theory of manifolds has developed with breathtaking speed.