## 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.

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.

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.

### 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.

What does Professor Bray do in differential geometry?

Professor Bray uses differential geometry to understand general relativity, and general relativity to motivate interesting problems in differential geometry.

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.